Program derivation — Coq 8.16.1 documentation

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# Program derivation

Coq comes with an extension called `Derive`, which supports program derivation. Typically in the style of Bird and Meertens or derivations of program refinements. To use the Derive extension it must first be required with `Require Coq.derive.Derive`. When the extension is loaded, it provides the following command:

Command Derive ident1 SuchThat one_term As ident2

  `ident1` can appear in `one_term`. This command opens a new proof presenting the user with a goal for `one_term` in which the name `ident1` is bound to an existential variable `?x` (formally, there are other goals standing for the existential variables but they are shelved, as described in `shelve`).

  When the proof ends two constants are defined:

  - The first one is named `ident1` and is defined as the proof of the shelved goal (which is also the value of `?x`). It is always transparent.

  - The second one is named `ident2`. It has type `type`, and its body is the proof of the initially visible goal. It is opaque if the proof ends with `Qed`, and transparent if the proof ends with `Defined`.

Example

Require Coq.derive.Derive.
  [Loading ML file derive_plugin.cmxs (using legacy method) ... done]
Require Import Coq.Numbers.Natural.Peano.NPeano.
Section P.
Variables (n m k: nat).
  n is declared m is declared k is declared
Derive p SuchThat ((k*n)+(k*m) = p) As h.
  1 focused goal (shelved: 1) n, m, k : nat p := ?Goal : nat ============================ k * n + k * m = p
Proof.
rewrite <- Nat.mul_add_distr_l.
  1 focused goal (shelved: 1) n, m, k : nat p := ?Goal : nat ============================ k * (n + m) = p
subst p.
  1 focused goal (shelved: 1) n, m, k : nat ============================ k * (n + m) = ?Goal
reflexivity.
  No more goals.
Qed.
End P.
Print p.
  p = fun n m k : nat => k * (n + m) : nat -> nat -> nat -> nat Arguments p (n m k)%nat_scope
Check h.
  h : forall n m k : nat, k * n + k * m = p n m k

Any property can be used as `term`, not only an equation. In particular, it could be an order relation specifying some form of program refinement or a non-executable property from which deriving a program is convenient.

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