Fixing tauto "special" behavior on singleton types w/ 2 parameters (bug #2680).

- tauto/intuition now works uniformly on and, prod, or, sum, False,
  Empty_set, unit, True (and isomorphic copies of them), iff, ->, and
  on all inhabited singleton types with a no-arguments constructor
  such as "eq t t" (even though the last case goes out of
  propositional logic: this features is so often used that it is
  difficult to come back on it).
- New dtauto and dintuition works on all inductive types with one
  constructors and no real arguments (for instance, they work on
  records such as "Equivalence"), in addition to -> and eq-like types.
- Moreover, both of them no longer unfold inner negations (this is a
  souce of incompatibility for intuition and evaluation of the level
  of incompatibility on contribs still needs to be done).

Incidentally, and amazingly, fixing bug #2680 made that constants
InfA_compat and InfA_eqA in SetoidList.v lost one argument: old tauto
had indeed destructed a section hypothesis "@StrictOrder A ltA@
thinking it was a conjunction, making this section hypothesis
artificially necessary while it was not.

Renouncing to the unfolding of inner negations made auto/eauto
sometimes succeeding more, sometimes succeeding less. There is by the
way a (standard) problem with not in auto/eauto: even when given as an
"unfold hint", it works only in goals, not in hypotheses, so that auto
is not able to solve something like "forall P, (forall x, ~ P x) -> P
0 -> False". Should we automatically add a lemma of type "HYPS -> A ->
False" in the hint database everytime a lemma ""HYPS -> ~A" is
declared (and "unfold not" is a hint), and similarly for all unfold
hints?

At this occasion, also re-did some proofs of Znumtheory.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15180 85f007b7-540e-0410-9357-904b9bb8a0f7

2 files changed, 4 insertions, 12 deletions

diff --git a/theories/FSets/FMapFacts.v b/theories/FSets/FMapFacts.v
index 0c1448c9b..9fef1dc63 100644
--- a/theories/FSets/FMapFacts.v
+++ b/theories/FSets/FMapFacts.v
@@ -519,7 +519,7 @@ Proof.
 intros. rewrite eq_option_alt. intro e.
 rewrite <- find_mapsto_iff, elements_mapsto_iff.
 unfold eqb.
-rewrite <- findA_NoDupA; intuition; try apply elements_3w; eauto.
+rewrite <- findA_NoDupA; dintuition; try apply elements_3w; eauto.
 Qed.
 
 Lemma elements_b : forall m x,
@@ -679,8 +679,8 @@ Add Parametric Morphism elt : (@Empty elt)
  with signature Equal ==> iff as Empty_m.
 Proof.
 unfold Empty; intros m m' Hm; intuition.
-rewrite <-Hm in H0; eauto.
-rewrite Hm in H0; eauto.
+rewrite <-Hm in H0; eapply H, H0.
+rewrite Hm in H0; eapply H, H0.
 Qed.
 
 Add Parametric Morphism elt : (@is_empty elt)
@@ -1872,13 +1872,7 @@ Module OrdProperties (M:S).
     add_mapsto_iff by (auto with *).
   unfold O.eqke, O.ltk; simpl.
   destruct (E.compare t0 x); intuition.
-  right; split; auto; ME.order.
-  ME.order.
-  elim H.
-  exists e0; apply MapsTo_1 with t0; auto.
-  right; right; split; auto; ME.order.
-  ME.order.
-  right; split; auto; ME.order.
+  elim H; exists e0; apply MapsTo_1 with t0; auto.
   Qed.
 
   Lemma elements_Add_Above : forall m m' x e,
diff --git a/theories/FSets/FSetProperties.v b/theories/FSets/FSetProperties.v
index 1bad80615..eec7196b7 100644
--- a/theories/FSets/FSetProperties.v
+++ b/theories/FSets/FSetProperties.v
@@ -995,8 +995,6 @@ Module OrdProperties (M:S).
    leb_1, gtb_1, (H0 a) by auto with *.
   intuition.
   destruct (E.compare a x); intuition.
-  right; right; split; auto with *.
-  ME.order.
   Qed.
 
   Definition Above x s := forall y, In y s -> E.lt y x.