# 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 -/
theorem mul_comm' {α} [CommSemigroup α] (x y : α) : x * y = y * x := CommSemigroup.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',
• CommSemigroup to AddCommSemigroup,
• x * y to x + y and y * x to y + x, and
• CommSemigroup.mul_comm' to AddCommSemigroup.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 QuotientGroup QuotientAddGroup


Later uses of to_additive on declarations in the QuotientGroup namespace will be created in the QuotientAddGroup 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 NewNamespace.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.

The to_additive_ignore_args attribute.

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The to_additive_relevant_arg attribute.

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The to_additive_reorder attribute.

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The to_additive_change_numeral attribute.

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An attr := ... option for to_additive.

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A reorder := ... option for to_additive.

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Options to to_additive.

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Options to to_additive.

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Remaining arguments of to_additive.

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The to_additive attribute.

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The to_additive attribute.

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A set of strings of names that end in a capital letter.

• If the string contains a lowercase letter, the string should be split between the first occurrence of a lower-case letter followed by an upper-case letter.
• If multiple strings have the same prefix, they should be grouped by prefix
• In this case, the second list should be prefix-free (no element can be a prefix of a later element)

Todo: automate the translation from String to an element in this RBMap (but this would require having something similar to the rb_lmap from Lean 3).

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partial def String.splitCase (s : String) (i₀ : ) (r : optParam () []) :

This function takes a String and splits it into separate parts based on the following (naming conventions)[https://github.com/leanprover-community/mathlib4/wiki#naming-convention].

E.g. #eval "InvHMulLEConjugate₂SMul_ne_top".splitCase yields ["Inv", "HMul", "LE", "Conjugate₂", "SMul", "_", "ne", "_", "top"].

Linter to check that the reorder attribute is not given manually

Linter, mostly used by @[to_additive], that checks that the source declaration doesn't have certain attributes

Linter to check that the to_additive attribute is not given manually

Linter to check whether the user correctly specified that the additive declaration already exists

An attribute that tells @[to_additive] that certain arguments of this definition are not involved when using @[to_additive]. This helps the heuristic of @[to_additive] by also transforming definitions if ℕ or another fixed type occurs as one of these arguments.

An attribute that stores all the declarations that needs their arguments reordered when applying @[to_additive]. It is applied automatically by the (reorder := ...) syntax of to_additive, and should not usually be added manually.

An attribute that is automatically added to declarations tagged with @[to_additive], if needed.

This attribute tells which argument is the type where this declaration uses the multiplicative structure. If there are multiple argument, we typically tag the first one. If this argument contains a fixed type, this declaration will note be additivized. See the Heuristics section of to_additive.attr for more details.

If a declaration is not tagged, it is presumed that the first argument is relevant. @[to_additive] uses the function to_additive.first_multiplicative_arg to automatically tag declarations. It is ok to update it manually if the automatic tagging made an error.

Implementation note: we only allow exactly 1 relevant argument, even though some declarations (like prod.group) have multiple arguments with a multiplicative structure on it. The reason is that whether we additivize a declaration is an all-or-nothing decision, and if we will not be able to additivize declarations that (e.g.) talk about multiplication on ℕ × α anyway.

Warning: interactions between this and the (reorder := ...) argument are not well-tested.

An attribute that stores all the declarations that deal with numeric literals on variable types.

Numeral literals occur in expressions without type information, so in order to decide whether 1 needs to be changed to 0, the context around the numeral is relevant. Most numerals will be in an OfNat.ofNat application, though tactics can add numeral literals inside arbitrary functions. By default we assume that we do not change numerals, unless it is in a function application with the to_additive_change_numeral attribute.

@[to_additive_change_numeral n₁ ...] should be added to all functions that take one or more numerals as argument that should be changed if additiveTest succeeds on the first argument, i.e. when the numeral is only translated if the first argument is a variable (or consists of variables). The arguments n₁ ... are the positions of the numeral arguments (starting counting from 1).

Maps multiplicative names to their additive counterparts.

Get the multiplicative → additive translation for the given name.

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def ToAdditive.insertTranslation (src : Lean.Name) (tgt : Lean.Name) (failIfExists : ) :

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Config is the type of the arguments that can be provided to to_additive.

• trace : Bool

View the trace of the to_additive procedure. Equivalent to set_option trace.to_additive true.

• tgt : Lean.Name

The name of the target (the additive declaration).

• doc :

An optional doc string.

• allowAutoName : Bool

If allowAutoName is false (default) then @[to_additive] will check whether the given name can be auto-generated.

• reorder : List ()

The arguments that should be reordered by to_additive, using cycle notation.

• attrs :

The attributes which we want to give to both the multiplicative and additive versions. For certain attributes (such as simp and simps) this will also add generated lemmas to the translation dictionary.

• The Syntax element corresponding to the original multiplicative declaration (or the to_additive attribute if it is added later), which we need for adding definition ranges.

• existing :

An optional flag stating whether the additive declaration already exists. If this flag is set but wrong about whether the additive declaration exists, to_additive will raise a linter error. Note: the linter will never raise an error for inductive types and structures.

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unsafe def ToAdditive.additiveTestUnsafe (findTranslation? : ) (ignore : Lean.NameOption ()) (e : Lean.Expr) :

Implementation function for additiveTest. We cache previous applications of the function, using the same method that Expr.find? uses, to avoid visiting the same subexpression many times. Note that we only need to cache the expressions without taking the value of inApp into account, since inApp only matters when the expression is a constant. However, for this reason we have to make sure that we never cache constant expressions, so that's why the ifs in the implementation are in this order.

Note that this function is still called many times by applyReplacementFun and we're not remembering the cache between these calls.

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unsafe def ToAdditive.additiveTestUnsafe.visit (findTranslation? : ) (ignore : Lean.NameOption ()) (e : Lean.Expr) (inApp : ) :
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def ToAdditive.additiveTest (findTranslation? : ) (ignore : Lean.NameOption ()) (e : Lean.Expr) :

additiveTest e tests whether the expression e contains a constant nm that is not applied to any arguments, and such that translations.find?[nm] = none. This is used in @[to_additive] for deciding which subexpressions to transform: we only transform constants if additiveTest applied to their first argument returns true. This means we will replace expression applied to e.g. α or α × β, but not when applied to e.g. ℕ or ℝ × α. We ignore all arguments specified by the ignore NameMap.

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def List.swapFirstTwo {α : Type u_1} :
List αList α

Swap the first two elements of a list

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Change the numeral nat_lit 1 to the numeral nat_lit 0. Leave all other expressions unchanged.

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applyReplacementFun e replaces the expression e with its additive counterpart. It translates each identifier (inductive type, defined function etc) in an expression, unless

• The identifier occurs in an application with first argument arg; and
• test arg is false. However, if f is in the dictionary relevant, then the argument relevant.find f is tested, instead of the first argument.

It will also reorder arguments of certain functions, using reorderFn: e.g. g x₁ x₂ x₃ ... xₙ becomes g x₂ x₁ x₃ ... xₙ if reorderFn g = some [1].

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def ToAdditive.applyReplacementFun.aux (findTranslation? : ) (reorderFn : Lean.NameList ()) (ignore : Lean.NameOption ()) (changeNumeral? : Lean.NameOption ()) (relevantArg : ) (trace : Bool) :

Implementation of applyReplacementFun.

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Eta expands e at most n times.

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e.expand eta-expands all expressions that have as head a constant n in reorder. They are expanded until they are applied to one more argument than the maximum in reorder.find n.

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def ToAdditive.reorderForall (src : Lean.Expr) (reorder : optParam (List ()) []) :

Reorder pi-binders. See doc of reorderAttr for the interpretation of the argument

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def ToAdditive.reorderLambda (src : Lean.Expr) (reorder : optParam (List ()) []) :

Reorder lambda-binders. See doc of reorderAttr for the interpretation of the argument

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def ToAdditive.updateDecl (tgt : Lean.Name) (srcDecl : Lean.ConstantInfo) (reorder : optParam (List ()) []) :

Run applyReplacementFun on the given srcDecl to make a new declaration with name tgt

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def ToAdditive.findTargetName (env : Lean.Environment) (src : Lean.Name) (pre : Lean.Name) (tgt_pre : Lean.Name) :

Find the target name of pre and all created auxiliary declarations.

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Returns a NameSet of all auxiliary constants in e that might have been generated when adding pre to the environment. Examples include pre.match_5, Mathlib.MyFile._auxLemma.3 and _private.Mathlib.MyFile.someOtherNamespace.someOtherDeclaration._eq_2. The last two examples may or may not have been generated by this declaration. The last example may or may not be the equation lemma of a declaration with the @[to_additive] attribute. We will only translate it has the @[to_additive] attribute.

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partial def ToAdditive.transformDeclAux (cfg : ToAdditive.Config) (pre : Lean.Name) (tgt_pre : Lean.Name) :

transform the declaration src and all declarations pre._proof_i occurring in src using the transforms dictionary. replace_all, trace, ignore and reorder are configuration options. pre is the declaration that got the @[to_additive] attribute and tgt_pre is the target of this declaration.

Copy the instance attribute in a to_additive

[todo] it seems not to work when the to_additive is added as an attribute later.

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def ToAdditive.warnExt {σ : Type} {α : Type} {β : Type} [] (stx : Lean.Syntax) (ext : ) (f : σ) (thisAttr : Lean.Name) (attrName : Lean.Name) (src : Lean.Name) (tgt : Lean.Name) :

Warn the user when the multiplicative declaration has an attribute.

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def ToAdditive.warnAttr {β : Type} {α : Type} [] (stx : Lean.Syntax) (attr : ) (f : β) (thisAttr : Lean.Name) (attrName : Lean.Name) (src : Lean.Name) (tgt : Lean.Name) :

Warn the user when the multiplicative declaration has a simple scoped attribute.

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def ToAdditive.warnParametricAttr {β : Type} (stx : Lean.Syntax) (attr : ) (thisAttr : Lean.Name) (attrName : Lean.Name) (src : Lean.Name) (tgt : Lean.Name) :

Warn the user when the multiplicative declaration has a parametric attribute.

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def ToAdditive.additivizeLemmas {m : } [] [] [] (names : ) (desc : String) (t : Lean.Namem ) :

runAndAdditivize names desc t runs t on all elements of names and adds translations between the generated lemmas (the output of t). names must be non-empty.

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Find the first argument of nm that has a multiplicative type-class on it. Returns 1 if there are no types with a multiplicative class as arguments. E.g. Prod.Group returns 1, and Pi.One returns 2. Note: we only consider the first argument of each type-class. E.g. [Pow A N] is a multiplicative type-class on A, not on N.

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partial def ToAdditive.capitalizeLikeAux (s : String) (i : ) (p : String) :

Helper for capitalizeLike.

Capitalizes s char-by-char like r. If s is longer, it leaves the tail untouched.

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Capitalize First element of a list like s. Note that we need to capitalize multiple characters in some cases, in examples like HMul or hAdd.

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Dictionary used by guessName to autogenerate names.

Note: guessName capitalizes first element of the output according to capitalization of the input. Input and first element should therefore be lower-case, 2nd element should be capitalized properly.

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Turn each element to lower-case, apply the nameDict and capitalize the output like the input.

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There are a few abbreviations we use. For example "Nonneg" instead of "ZeroLE" or "addComm" instead of "commAdd". Note: The input to this function is case sensitive! Todo: A lot of abbreviations here are manual fixes and there might be room to improve the naming logic to reduce the size of fixAbbreviation.

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Autogenerate additive name. This runs in several steps:

1. Split according to capitalisation rule and at _.
2. Apply word-by-word translation rules.
3. Fix up abbreviations that are not word-by-word translations, like "addComm" or "Nonneg".
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Return the provided target name or autogenerate one if one was not provided.

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def ToAdditive.proceedFieldsAux (src : Lean.Name) (tgt : Lean.Name) (f : ) :

if f src = #[a_1, ..., a_n] and f tgt = #[b_1, ... b_n] then proceedFieldsAux src tgt f will insert translations from src.a_i to tgt.b_i.

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Add the structure fields of src to the translations dictionary so that future uses of to_additive will map them to the corresponding tgt fields.

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Elaboration of the configuration options for to_additive.

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partial def ToAdditive.applyAttributes (stx : Lean.Syntax) (rawAttrs : ) (thisAttr : Lean.Name) (src : Lean.Name) (tgt : Lean.Name) :

Apply attributes to the multiplicative and additive declarations.

Copies equation lemmas and attributes from src to tgt
Make a new copy of a declaration, replacing fragments of the names of identifiers in the type and the body using the translations dictionary. This is used to implement @[to_additive].
addToAdditiveAttr src cfg adds a @[to_additive] attribute to src with configuration cfg. See the attribute implementation for more details. It returns an array with names of additive declarations (usually 1, but more if there are nested to_additive` calls.