The `@[to_additive]` attribute. #

The attribute to_additive can be used to automatically transport theorems and definitions (but not inductive types and structures) from a multiplicative theory to an additive theory.

To use this attribute, just write:

@[to_additive]
theorem mul_comm' {α} [CommSemigroup α] (x y : α) : x * y = y * x := mul_comm x y

This code will generate a theorem named add_comm'. It is also possible to manually specify the name of the new declaration:

@[to_additive add_foo]
theorem foo := sorry

An existing documentation string will not be automatically used, so if the theorem or definition has a doc string, a doc string for the additive version should be passed explicitly to to_additive.

/-- Multiplication is commutative -/
@[to_additive "Addition is commutative"]
theorem mul_comm' {α} [comm_semigroup α] (x y : α) : x * y = y * x := comm_semigroup.mul_comm

The transport tries to do the right thing in most cases using several heuristics described below. However, in some cases it fails, and requires manual intervention.

Use the (reorder := ...) syntax to reorder the arguments in the generated additive declaration. This is specified using cycle notation. For example (reorder := 1 2, 5 6) swaps the first two arguments with each other and the fifth and the sixth argument and (reorder := 3 4 5) will move the fifth argument before the third argument. This is mostly useful to translate declarations using Pow to those using SMul.

Use the (attr := ...) syntax to apply attributes to both the multiplicative and the additive version:

@[to_additive (attr := simp)] lemma mul_one' {G : Type*} [group G] (x : G) : x * 1 = x := mul_one x

For simp and simps this also ensures that some generated lemmas are added to the additive dictionary. @[to_additive (attr := to_additive)] is a special case, where the to_additive attribute is added to the generated lemma only, to additivize it again. This is useful for lemmas about Pow to generate both lemmas about SMul and VAdd. Example:

@[to_additive (attr := to_additive VAdd_lemma, simp) SMul_lemma]
lemma Pow_lemma ... :=

In the above example, the simp is added to all 3 lemmas. All other options to to_additive (like the generated name or (reorder := ...)) are not passed down, and can be given manually to each individual to_additive call.

Implementation notes #

The transport process generally works by taking all the names of identifiers appearing in the name, type, and body of a declaration and creating a new declaration by mapping those names to additive versions using a simple string-based dictionary and also using all declarations that have previously been labeled with to_additive.

In the mul_comm' example above, to_additive maps:

mul_comm' to add_comm',
comm_semigroup to add_comm_semigroup,
x * y to x + y and y * x to y + x, and
comm_semigroup.mul_comm' to add_comm_semigroup.add_comm'.

Heuristics #

to_additive uses heuristics to determine whether a particular identifier has to be mapped to its additive version. The basic heuristic is

Only map an identifier to its additive version if its first argument doesn't contain any unapplied identifiers.

Examples:

@Mul.mul Nat n m (i.e. (n * m : Nat)) will not change to +, since its first argument is Nat, an identifier not applied to any arguments.
@Mul.mul (α × β) x y will change to +. It's first argument contains only the identifier prod, but this is applied to arguments, α and β.
@Mul.mul (α × Int) x y will not change to +, since its first argument contains Int.

The reasoning behind the heuristic is that the first argument is the type which is "additivized", and this usually doesn't make sense if this is on a fixed type.

There are some exceptions to this heuristic:

Identifiers that have the @[to_additive] attribute are ignored. For example, multiplication in ↥Semigroup is replaced by addition in ↥AddSemigroup.
If an identifier d has attribute @[to_additive_relevant_arg n] then the argument in position n is checked for a fixed type, instead of checking the first argument. @[to_additive] will automatically add the attribute @[to_additive_relevant_arg n] to a declaration when the first argument has no multiplicative type-class, but argument n does.
If an identifier has attribute @[to_additive_ignore_args n1 n2 ...] then all the arguments in positions n1, n2, ... will not be checked for unapplied identifiers (start counting from 1). For example, ContMDiffMap has attribute @[to_additive_ignore_args 21], which means that its 21st argument (n : WithTop ℕ) can contain ℕ (usually in the form Top.top ℕ ...) and still be additivized. So @Mul.mul (C^∞⟮I, N; I', G⟯) _ f g will be additivized.

Troubleshooting #

If @[to_additive] fails because the additive declaration raises a type mismatch, there are various things you can try. The first thing to do is to figure out what @[to_additive] did wrong by looking at the type mismatch error.

Option 1: The most common case is that it didn't additivize a declaration that should be additivized. This happened because the heuristic applied, and the first argument contains a fixed type, like ℕ or ℝ. However, the heuristic misfires on some other declarations. Solutions:
- First figure out what the fixed type is in the first argument of the declaration that didn't get additivized. Note that this fixed type can occur in implicit arguments. If manually finding it is hard, you can run set_option trace.to_additive_detail true and search the output for the fragment "contains the fixed type" to find what the fixed type is.
- If the fixed type has an additive counterpart (like ↥Semigroup), give it the @[to_additive] attribute.
- If the fixed type has nothing to do with algebraic operations (like TopCat), add the attribute @[to_additive existing Foo] to the fixed type Foo.
- If the fixed type occurs inside the k-th argument of a declaration d, and the k-th argument is not connected to the multiplicative structure on d, consider adding attribute [to_additive_ignore_args k] to d. Example: ContMDiffMap ignores the argument (n : WithTop ℕ)
Option 2: It additivized a declaration d that should remain multiplicative. Solution:
- Make sure the first argument of d is a type with a multiplicative structure. If not, can you reorder the (implicit) arguments of d so that the first argument becomes a type with a multiplicative structure (and not some indexing type)? The reason is that @[to_additive] doesn't additivize declarations if their first argument contains fixed types like ℕ or ℝ. See section Heuristics. If the first argument is not the argument with a multiplicative type-class, @[to_additive] should have automatically added the attribute @[to_additive_relevant_arg] to the declaration. You can test this by running the following (where d is the full name of the declaration):
```
  open Lean in run_cmd logInfo m!"{ToAdditive.relevantArgAttr.find? (← getEnv) `d}"
```
  The expected output is n where the n-th (0-indexed) argument of d is a type (family) with a multiplicative structure on it. none means 0. If you get a different output (or a failure), you could add the attribute @[to_additive_relevant_arg n] manually, where n is an (1-indexed) argument with a multiplicative structure.
Option 3: Arguments / universe levels are incorrectly ordered in the additive version. This likely only happens when the multiplicative declaration involves pow/^. Solutions:
- Ensure that the order of arguments of all relevant declarations are the same for the multiplicative and additive version. This might mean that arguments have an "unnatural" order (e.g. Monoid.npow n x corresponds to x ^ n, but it is convenient that Monoid.npow has this argument order, since it matches AddMonoid.nsmul n x.
- If this is not possible, add (reorder := ...) argument to to_additive.

If neither of these solutions work, and to_additive is unable to automatically generate the additive version of a declaration, manually write and prove the additive version. Often the proof of a lemma/theorem can just be the multiplicative version of the lemma applied to multiplicative G. Afterwards, apply the attribute manually:

attribute [to_additive foo_add_bar] foo_bar

This will allow future uses of to_additive to recognize that foo_bar should be replaced with foo_add_bar.

Handling of hidden definitions #

Before transporting the “main” declaration src, to_additive first scans its type and value for names starting with src, and transports them. This includes auxiliary definitions like src._match_1, src._proof_1.

In addition to transporting the “main” declaration, to_additive transports its equational lemmas and tags them as equational lemmas for the new declaration.

Structure fields and constructors #

If src is a structure, then the additive version has to be already written manually. In this case to_additive adds all structure fields to its mapping.

Name generation #

If @[to_additive] is called without a name argument, then the new name is autogenerated. First, it takes the longest prefix of the source name that is already known to to_additive, and replaces this prefix with its additive counterpart. Second, it takes the last part of the name (i.e., after the last dot), and replaces common name parts (“mul”, “one”, “inv”, “prod”) with their additive versions.
[todo] Namespaces can be transformed using map_namespace. For example:
```
run_cmd to_additive.map_namespace `quotient_group `quotient_add_group
```
Later uses of to_additive on declarations in the quotient_group namespace will be created in the quotient_add_group namespaces.
If @[to_additive] is called with a name argument new_name /without a dot/, then to_additive updates the prefix as described above, then replaces the last part of the name with new_name.
If @[to_additive] is called with a name argument new_namespace.new_name /with a dot/, then to_additive uses this new name as is.

As a safety check, in the first case to_additive double checks that the new name differs from the original one.