Chapter 22  Program

• Elaborating programs
• Solving obligations
• Frequently Asked Questions

Matthieu Sozeau

We present here the new Program tactic commands, used to build certified Coq programs, elaborating them from their algorithmic skeleton and a rich specification [125]. It can be sought of as a dual of extraction (see Chapter 21). The goal of Program is to program as in a regular functional programming language whilst using as rich a specification as desired and proving that the code meets the specification using the whole Coq proof apparatus. This is done using a technique originating from the “Predicate subtyping” mechanism of PVS[122], which generates type-checking conditions while typing a term constrained to a particular type. Here we insert existential variables in the term, which must be filled with proofs to get a complete Coq term. Program replaces the Program tactic by Catherine Parent [111] which had a similar goal but is no longer maintained.

The languages available as input are currently restricted to Coq’s term language, but may be extended to Objective Caml, Haskell and others in the future. We use the same syntax as Coq and permit to use implicit arguments and the existing coercion mechanism. Input terms and types are typed in an extended system (Russell) and interpreted into Coq terms. The interpretation process may produce some proof obligations which need to be resolved to create the final term.

22.1  Elaborating programs

The main difference from Coq is that an object in a type T : Set can be considered as an object of type { x : T | P} for any wellformed P : Prop. If we go from T to the subset of T verifying property P, we must prove that the object under consideration verifies it. Russell will generate an obligation for every such coercion. In the other direction, Russell will automatically insert a projection.

Another distinction is the treatment of pattern-matching. Apart from the following differences, it is equivalent to the standard match operation (see Section 4.5.4).

• Generation of equalities. A match expression is always generalized by the corresponding equality. As an example, the expression:
  Coq <   match x with
  Coq <   | 0 => t
  Coq <   | S n => u
  Coq <   end.
  

  will be first rewrote to:

  Coq <   (match x as y return (x = y -> _) with
  Coq <   | 0 => fun H : x = 0 -> t
  Coq <   | S n => fun H : x = S n -> u
  Coq <   end) (refl_equal n).
  

  This permits to get the proper equalities in the context of proof obligations inside clauses, without which reasoning is very limited.

• Generation of inequalities. If a pattern intersects with a previous one, an inequality is added in the context of the second branch. See for example the definition of div2 below, where the second branch is typed in a context where ∀ p, _ <> S (S p).
• Coercion. If the object being matched is coercible to an inductive type, the corresponding coercion will be automatically inserted. This also works with the previous mechanism.

If you do specify a return or in clause the typechecker will fall back directly to Coq’s usual typing of dependent pattern-matching.

The next two commands are similar to their standard counterparts Definition (see Section 1.3.2) and Fixpoint (see Section 1.3.4) in that they define constants. However, they may require the user to prove some goals to construct the final definitions.

22.1.1  Program Definition ident := term.

This command types the value term in Russell and generate proof obligations. Once solved using the commands shown below, it binds the final Coq term to the name ident in the environment.

Error messages:

1. ident already exists

Variants:

1. Program Definition ident :term₁ := term₂.
   It interprets the type term₁, potentially generating proof obligations to be resolved. Once done with them, we have a Coq type term₁′. It then checks that the type of the interpretation of term₂ is coercible to term₁′, and registers ident as being of type term₁′ once the set of obligations generated during the interpretation of term₂ and the aforementioned coercion derivation are solved.
2. Program Definition ident binder₁…binder_n :term₁ := term₂.
   This is equivalent to
   Program Definition ident : forall binder₁…binder_n, term₁ :=
       fun binder₁…binder_n => term₂ .

Error messages:

1. In environment … the term: term₂ does not have type term₁.
   Actually, it has type term₃.

See also: Sections 6.9.1, 6.9.2, 8.5.5

22.1.2  Program Fixpoint ident params {order} : type := term

The structural fixpoint operator behaves just like the one of Coq (see Section 1.3.4), except it may also generate obligations. It works with mutually recursive definitions too.

Here we have one obligation for each branch (branches for 0 and (S 0) are automatically generated by the pattern-matching compilation algorithm).

One can use a well-founded order or a measure as termination orders using the syntax:

The measure keyword expects a measure function into the naturals, whereas wf expects a relation.

Caution

When defining structurally recursive functions, the generated obligations should have the prototype of the currently defined functional in their context. In this case, the obligations should be transparent (e.g. using Defined) so that the guardedness condition on recursive calls can be checked by the kernel’s type-checker. There is an optimization in the generation of obligations which gets rid of the hypothesis corresponding to the functionnal when it is not necessary, so that the obligation can be declared opaque (e.g. using Qed). However, as soon as it appears in the context, the proof of the obligation is required to be declared transparent.

No such problems arise when using measures or well-founded recursion.

An important point about well-founded and measure-based functions is the following: The recursive prototype of a function of type binder₁…binder_n { measure m binder_i }:type₁, inside the body is {binder_i′ | m x_i′ < m x_i}…binder_n,type₁. So any arguments appearing before the recursive one are ignored for the recursive calls, hence they are constant.

22.1.3  Program Lemma ident : type.

The Russell language can also be used to type statements of logical properties. It will generate obligations, try to solve them automatically and fail if some unsolved obligations remain. In this case, one can first define the lemma’s statement using Program Definition and use it as the goal afterwards. Otherwise the proof will be started with the elobarted version as a goal. The Program prefix can similarly be used as a prefix for Variable, Hypothesis, Axiom etc...

22.2  Solving obligations

The following commands are available to manipulate obligations. The optional identifier is used when multiple functions have unsolved obligations (e.g. when defining mutually recursive blocks). The optional tactic is replaced by the default one if not specified.

• Obligation Tactic := expr Sets the default obligation solving tactic applied to all obligations automatically, whether to solve them or when starting to prove one, e.g. using Next.
• Obligations [of ident] Displays all remaining obligations.
• Obligation num [of ident] Start the proof of obligation num.
• Next Obligation [of ident] Start the proof of the next unsolved obligation.
• Solve Obligations [of ident] [using expr] Tries to solve each obligation of identusing the given tactic or the default one.
• Solve All Obligations [using expr] Tries to solve each obligation of every program using the given tactic or the default one (useful for mutually recursive definitions).
• Admit Obligations [of ident] Admits all obligations (does not work with structurally recursive programs).
• Preterm [of ident] Shows the term that will be fed to the kernel once the obligations are solved. Useful for debugging.
• Set Transparent Obligations Control whether all obligations should be declared as transparent (the default), or if the system should infer which obligations can be declared opaque.

The module Coq.Program.Tactics defines the default tactic for solving obligations called program_simpl. Importing Coq.Program.Program also adds some useful notations, as documented in the file itself.

22.3  Frequently Asked Questions

• Ill-formed recursive definitions This error can happen when one tries to define a function by structural recursion on a subset object, which means the Coq function looks like:

  Program Fixpoint f (x : A | P) := match x with A b => f b end.

  Supposing b : A, the argument at the recursive call to f is not a direct subterm of x as b is wrapped inside an exist constructor to build an object of type {x : A | P}. Hence the definition is rejected by the guardedness condition checker. However you can do wellfounded recursion on subset objects like this:

  Program Fixpoint f (x : A | P) { measure size } :=
    match x with A b => f b end.
  

  You will then just have to prove that the measure decreases at each recursive call. There are three drawbacks though:

  1. You have to define the measure yourself;
  2. The reduction is a little more involved, although it works using lazy evaluation;
  3. Mutual recursion on the underlying inductive type isn’t possible anymore, but nested mutual recursion is always possible.

• webmaster
• xhtml valid
• CSS valid