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Class instances for Coq inductive types with little boilerplate


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Deriving ─ Generic instances for Coq inductive types


The Deriving library builds instances of basic MathComp classes for inductive data types with little boilerplate, akin to Haskell's deriving functionality. To define an eqType instance for a type foo, just write:

From HB Require Import structures.
From mathcomp Require Import ssreflect ssrnat eqtype.
From deriving Require Import deriving.

Inductive foo := Foo1 of nat | Foo2 of foo & nat.

(* foo_rect is the recursor automatically generated by Coq *)
Definition foo_indDef := [indDef for foo_rect].
Canonical foo_indType := IndType foo foo_indDef.

Definition foo_hasDecEq := [derive hasDecEq for foo].
HB.instance Definition _ := foo_hasDecEq.

Supported types and limitations

Besides simple definitions such as the one above, Deriving can handle the following features:

  • Types with uniform parameters (e.g. list).

  • Mutually inductive types (by using the recursor generated by Combined Scheme command).

  • Nested inductive types (if you write your own recursor).

Check the tests/ directory for detailed examples.

The following features are still not supported:

  • Types with non-uniform parameters (e.g. Vector.t).

  • Constructors with dependent types (e.g. C : forall n : nat, P n -> T).

  • Coinductive types.

Predefined instances

Besides eqType, there are predefined generic instances for choiceType, countType, finType and orderType. To use them, you must ensure that every non-recursive argument of the type is also an instance of the class; otherwise, you'll get an ugly, uninformative error message. For finType, you must additionally ensure that the type does not have recursive occurrences.

You can also define instances for your own classes inside of Coq, without resorting to OCaml code. This feature is not documented yet, but you can refer to the definition of the instances provided by Deriving in theories/instances. Or drop me a line!

Record instances

Coq does not generate induction principles for record types by default. If you want to derive an instance for a record type, you need to generate the induction principle by hand:

Record foo := (* ... *)
Scheme foo_rect := Induction for foo Sort Type.

Check the tests for an example.

Simplification and performance issues

It is useful for instances of certain classes to have good reduction behavior (e.g. eqType). By default, Deriving attempts to simplify the derived instances as much as possible, to make them more similar to hand-written code. However, this process can be too slow for large definitions, so it can be disabled with the nored flag:

Definition large_type_hasDecEq : Equality.mixin_of large_type.
Proof. exact [derive nored hasDecEq for large_type]. Qed.

In such cases, it is a good idea to keep the instance opaque (e.g. defined with Qed) to avoid slowdown.


  • Coq 8.17 -- 8.18

  • coq-mathcomp-ssreflect 2.0.0


Deriving can be installed through the released repository:

opam repo add coq-released
opam install coq-deriving

Alternatively, you can compile Deriving from source:

make && make install


If you want to cite Deriving, you can refer to its Zenodo page.


Class instances for Coq inductive types with little boilerplate







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