The Coq mailing list

[Matthieu Sozeau <mattam AT mattam.org>]

Le 17 sept. 08 à 20:28, Brian Aydemir a écrit :

> Hi,

Hi,

> At the end of this message, I include a snippet of code that  
> illustrates a situation in which I am unable to simplify a function  
> defined using Program Fixpoint.  I'm using the subversion version of  
> Coq 8.2 (r11417).  The code is a simplified (somewhat silly) version  
> of what I actually want to do, but hopefully it's not too trivial in  
> this form.  I have noted in comments variations that I've tried, to  
> no success.
>
> The main function of interest is [silly_subst], which is defined  
> using [size] as a measure.  I am unable to get [silly_subst] to  
> simplify in the proof of [silly_subst_thm].  Any tips would be  
> appreciated.

It can be indeed quite difficult to see where the computations
get stuck in this situation.

> (* too lazy to prove this right now *)
> (* as noted below, I don't actually need this lemma for the problem  
> to appear *)
> Axiom lc_rename : forall x y e,
>  lc (open_rec 0 (fvar x) e) ->
>  lc (open_rec 0 (fvar y) e).

That's why it blocks, at the recursive call in the [lc_abs] case,
you give this axiom which makes reduction stop right after, on a
[match (lc_rename e x0 7 pf) with ...]. You can see that by doing
some unfolding and simplification (and wait a bit because the
resulting term is huge).

I've developped two tactics that try to fold/unfold such
definitions based on 
[@Fix_measure_sub],
 avoiding bloated goals.
For some reason I never put them in [FunctionalExtensionality] :)
The [fold_sub] tactic is only using convertibility to fold a
definition but the [unfold_sub] rewrites by a lemma using
extensionality. The calling convention is a bit weird: you have
to give the function and the function applied to the arguments
that are constant during the recursive calls to [unfold_sub] and
only the latter to [fold_sub].

<<
Ltac fold_sub f :=
  match goal with
    | [ |- ?T ] =>
      match T with
        appcontext C [ @Fix_measure_sub _ _ _ _ ?arg ] =>
        let app := context C [ f arg ] in
         change app
      end
  end.

Ltac unfold_sub f fargs :=
  set (call:=fargs) ; unfold f in call ; unfold call ; clear call ;
  rewrite fix_sub_measure_eq_ext ; repeat fold_sub fargs ; simpl  
proj1_sig.
>>

Using these, you can avoid seeing the unfolding to [Fix_measure_sub]
while doing one unfolding. For example, at the end of your script you  
get:

<<
Lemma silly_subst_thm : forall (x : nat) (u e : exp) (pf : lc e),
 silly_subst x u e pf = e.
Proof.

 induction pf; intros; simpl in *; program_simpl.
 (* Why did the first subgoal simplify away? <- program_simpl tries  
reflexivity *)
 (* Why aren't the remaining subgoals simplified or reduced in any  
way? *)
 unfold_sub silly_subst (silly_subst x u).

2 subgoals

  x : nat
  u : exp
  x0 : nat
  e : exp
  pf : lc (open_rec 0 (fvar x0) e)
  IHpf : silly_subst x u (open_rec 0 (fvar x0) e) pf = open_rec 0  
(fvar x0) e
  ============================
   silly_subst x u (open_rec 0 (fvar 7) e) (lc_rename e x0 7 pf) =  
abs e

subgoal 2 is:
 silly_subst x u (app e1 e2) (lc_app e1 e2 pf1 pf2) = app e1 e2
>>

If you try again to reduce you'll see that [lc_rename] comes back in  
head position.

Hope this helps,
-- Matthieu