Inference of return clause: giving uniformly priority to "small inversion".

As noted by Jason Gross on coq-club (Aug 18, 2016), the "small
inversion" heuristic is not used consistently depending on whether the
variables in the type constraint are Rel or Var.

This commit simply gives uniformly preference to the inversion of the
predicate along the indices of the type over other heuristics.

The next three commits will improve further a uniform use of the
different heuristics.

----------------------------------------------------------------------
Here are some extra comments on how to go further with the inference
of the return predicate:

The "small inversion" heuristic build_inversion_problem (1) is
characterized by two features:

- small inversion properly speaking (a), i.e. that is for a match on
  t:I params p1(u11..u1p1) ... pn(un1..unpn) with pi exposing the
  constructor structure of the indices of the type of t, a return
  clause of the form "fun x1..xn (y:I params x1..xn) => match x1..xn y with
  | p1(z11..z1p1) ... pn(zn1..znpn) => ?T@{z11..znpn}
  | _ => IDProp
  end" is used,

- the dependent subterms in the external type constraint U are replaced
  by existential variables (b) which can be filled either by projecting
  (i.e. installing a dependency) or imitating (i.e. no dependency);
  this is obtained by solving the constraint ?T@{u11..unpn} == U by
  setting ?T@{z11..znpn} := U'(...?wij@{zij:=uij}...) where U has been
  written under the form U'(...uij...) highlighting all occurrences of
  each of the uij occurring in U; otherwise said the problem is reduced to
  the question of instantiating each wij, deciding whether wij@{zij} := zij
  (projection) or wij@{zij} := uij (imitation) [There may be different
  way to expose the uij in U, e.g. in the presence of overlapping, or of
  evars in U; this is left undetermined].

The two other heuristics used are:

- prepare_predicate_from_arsign_tycon (2): takes the external type
  constraint U and decides that each subterm of the form xi or y for a
  match on "y:I params x1 ... xn" is dependent; otherwise said, it
  corresponds to the degenerated form of (1) where
  - no constructor structure is exposed (i.e. each pi is trivial)
  - only uij that are Rel are replaced by an evar ?wij and this evar is
    directly instantiated by projection (hence creating a dependency),

- simple use of of an evar in case no type constraint is given (3):
  this evar is not dependent on the indices nor on the term to match.

Heuristic (1) is not strictly more powerful than other heuristics
because of (at least) two weaknesses.

- The first weakness is due to feature (b), i.e. to letting
  unification decide whether these evars have to create a dependency
  (projection) or not (imitation).

  In particular, the heuristic (2) gives priority to systematic
  abstraction over the dependencies (i.e. giving priority to
  projection over imitation) and it can then be better as the
  following example (from RelationClasses.v) shows:

  Fixpoint arrows (l : Tlist) (r : Type) : Type :=
    match l with
    | Tnil => r
    | A :: l' => A -> arrows l' r
    end.

  Fixpoint predicate_all (l : Tlist) : arrows l Prop -> Prop :=
    match l with
    | Tnil => fun f => f
    | A :: tl => fun f => forall x : A, predicate_all tl (f x)
    end.

  Using (1) fails. It proposes the predicate
  "fun l' => arrows ?l[l':=l'] Prop" so that typing the first branch
  leads to unify "arrows ?l[l:=Tnil] Prop == Prop", a problem about
  which evarconv unification is not able (yet!) to see what are the
  two possible solutions. Using (2) works. It instead directly
  suggests that the predicate is "fun l => arrows l Prop" is used, so
  that unification is not needed.

  Even if in practice the (2) is good (and hence could be added to
  (1)), it is not universally better. Consider e.g.

     y:bool,H1:P y,H2:P y,f:forall y, P y -> Q y |-
              match y as z return Q y with
              | true  => f y H1
              | false => f y H2
              end : Q y

  There is no way to type it with clause "as z return Q z" even if
  trying to generalize H1 and H2 so that they get type P z.

- A second weakness is due to the interaction between small inversion
  and constructors having a type whose indices havex a less refined
  constructor structure than in the term to match, as in:

  Inductive I : nat -> Set :=
  | C1 : forall n : nat, listn n -> I n
  | C2 : forall n : nat, I n -> I n.

  Check (fun x : I 0 => match x with
                        | C1 n l => 0
                        | C2 n c => 0
                        end).

  where the inverted predicate is "in I n return match n with 0 => ?T | _ => IDProp end"
  but neither C1 nor C2 have fine enough types so that n becomes
  constructed. There is a generic solution to that kind of situation which
  is to compile the above into

  Check (fun x : I 0 => match x with
                        | C1 n l => match n with 0 => 0 | _ -> id end
                        | C2 n c => match n with 0 => 0 | _ -> id end
                        end).

  but this is not implemented yet.

  In the absence of this refinement, heuristic (3) can here work
  better.

So, the current status of the claim is that for (1) to be strictly
more powerful than other current heuristics, work has to be done

- (A) at the unification level (by either being able to reduce problems of
  the form "match ?x[constructor] with ... end = a-rigid-term", or, at
  worst, by being able to use the heuristic favoring projecting for such
  a problem), so that it is better than (2),

- (B) at the match compilation level, by enforcing that, in each branch,
  the corresponding constructor is refined so has to match (or
  discriminate) the constraints given by the type of the term to
  match, and hence being better than (3).

Moreover, (2) and (3) are disjoint. Here is an example which (3) can
solve but not (2) (and (1) cannot because of (B)). [To be fixed in
next commit.]

Inductive I : bool -> bool -> Type := C : I true true | D x : I x x.

Check fun z P Q (y:I true z) (H1 H2:P y) (f:forall y, P y -> Q y z) =>
              match y with
              | C  => f y H1
              | D _ => f y H2
              end : Q y z.

Indeed, (2) infers "as y' in I b z return Q y z" which does not work.

Here is an example which (2) can solve but not (3) (and (1) cannot
because of (B) again). [To be fixed in 2nd next commit].

Check fun z P Q (y:I true z) (H1 H2:P y) (f:forall y z, P y -> Q y z) =>
              match y with
              | C  => f y true H1
              | D b => f y b H2
              end : Q y z.

fix

1 files changed, 9 insertions, 13 deletions

diff --git a/pretyping/cases.ml b/pretyping/cases.ml
index 33ee579eca..a26e1830d7 100644
--- a/pretyping/cases.ml
+++ b/pretyping/cases.ml
@@ -1996,10 +1996,7 @@ let prepare_predicate_from_arsign_tycon env sigma loc tomatchs arsign c =
  * type and 1 assumption for each term not _syntactically_ in an
  * inductive type.
 
- * Each matched terms are independently considered dependent or not.
-
- * A type constraint but no annotation case: we try to specialize the
- * tycon to make the predicate if it is not closed.
+ * Each matched term is independently considered dependent or not.
  *)
 
 exception LocalOccur
@@ -2031,15 +2028,14 @@ let prepare_predicate ?loc typing_fun env sigma tomatchs arsign tycon pred =
     | None, Some t when not (noccur_with_meta sigma 0 max_int t) ->
 	(* If the tycon is not closed w.r.t real variables, we try *)
         (* two different strategies *)
-       (* First strategy: we abstract the tycon wrt to the dependencies *)
-         let sigma, t = refresh_tycon sigma t in
-         let p1 =
+        (* First strategy: we build an "inversion" predicate *)
+        let sigma1,pred1 = build_inversion_problem loc env sigma tomatchs t in
+        (* Optional second strategy: we abstract the tycon wrt to the dependencies *)
+        let p2 =
           prepare_predicate_from_arsign_tycon env sigma loc tomatchs arsign t in
-	(* Second strategy: we build an "inversion" predicate *)
-	let sigma2,pred2 = build_inversion_problem loc env sigma tomatchs t in
-	(match p1 with
-         | Some (sigma1,pred1,arsign) -> [sigma1, pred1, arsign; sigma2, pred2, arsign]
-         | None -> [sigma2, pred2, arsign])
+        (match p2 with
+         | Some (sigma2,pred2,arsign) -> [sigma1, pred1, arsign; sigma2, pred2, arsign]
+         | None -> [sigma1, pred1, arsign])
     | None, _ ->
 	(* No dependent type constraint, or no constraints at all: *)
 	(* we use two strategies *)
@@ -2052,7 +2048,7 @@ let prepare_predicate ?loc typing_fun env sigma tomatchs arsign tycon pred =
 	in
         (* First strategy: we build an "inversion" predicate *)
 	let sigma1,pred1 = build_inversion_problem loc env sigma tomatchs t in
-	(* Second strategy: we directly use the evar as a non dependent pred *)
+        (* Second strategy: we use the evar or tycon as a non dependent pred *)
         let pred2 = lift (List.length (List.flatten arsign)) t in
         [sigma1, pred1, arsign; sigma, pred2, arsign]
     (* Some type annotation *)