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

9 files changed, 38 insertions, 46 deletions

diff --git a/theories/Classes/EquivDec.v b/theories/Classes/EquivDec.v
index 9f44d4fef..87f86e0d3 100644
--- a/theories/Classes/EquivDec.v
+++ b/theories/Classes/EquivDec.v
@@ -139,7 +139,7 @@ Program Instance list_eqdec `(eqa : EqDec A eq) : ! EqDec (list A) eq :=
       | _, _ => in_right
     end }.
 
-  Next Obligation. destruct y ; intuition eauto. Defined.
+  Next Obligation. destruct y ; unfold not in *; eauto. Defined.
 
   Solve Obligations with unfold equiv, complement in * ; 
     program_simpl ; intuition (discriminate || eauto).
diff --git a/theories/Classes/RelationClasses.v b/theories/Classes/RelationClasses.v
index a8de1ba08..3717e1cb4 100644
--- a/theories/Classes/RelationClasses.v
+++ b/theories/Classes/RelationClasses.v
@@ -130,7 +130,7 @@ Tactic Notation "apply" "*" constr(t) :=
 
 Ltac simpl_relation :=
   unfold flip, impl, arrow ; try reduce ; program_simpl ;
-    try ( solve [ intuition ]).
+    try ( solve [ dintuition ]).
 
 Local Obligation Tactic := simpl_relation.
 
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.
diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v
index 70f4ff0d9..944e7da21 100644
--- a/theories/Reals/RIneq.v
+++ b/theories/Reals/RIneq.v
@@ -43,7 +43,7 @@ Hint Immediate Rge_refl: rorders.
 
 Lemma Rlt_irrefl : forall r, ~ r < r.
 Proof.
-  generalize Rlt_asym. intuition eauto.
+  intros r H; eapply Rlt_asym; eauto.
 Qed.
 Hint Resolve Rlt_irrefl: real.
 
@@ -64,7 +64,9 @@ Qed.
 (**********)
 Lemma Rlt_dichotomy_converse : forall r1 r2, r1 < r2 \/ r1 > r2 -> r1 <> r2.
 Proof.
-  generalize Rlt_not_eq Rgt_not_eq. intuition eauto.
+  intuition.
+  - apply Rlt_not_eq in H1. eauto.
+  - apply Rgt_not_eq in H1. eauto.
 Qed.
 Hint Resolve Rlt_dichotomy_converse: real.
 
@@ -74,7 +76,7 @@ Hint Resolve Rlt_dichotomy_converse: real.
 Lemma Req_dec : forall r1 r2, r1 = r2 \/ r1 <> r2.
 Proof.
   intros; generalize (total_order_T r1 r2) Rlt_dichotomy_converse;
-    intuition eauto 3.
+    unfold not; intuition eauto 3.
 Qed.
 Hint Resolve Req_dec: real.
 
@@ -175,7 +177,7 @@ Proof. eauto using Rnot_gt_ge with rorders. Qed.
 Lemma Rlt_not_le : forall r1 r2, r2 < r1 -> ~ r1 <= r2.
 Proof.
   generalize Rlt_asym Rlt_dichotomy_converse; unfold Rle in |- *.
-  intuition eauto 3.
+  unfold not; intuition eauto 3.
 Qed.
 Hint Immediate Rlt_not_le: real.
 
diff --git a/theories/Reals/ROrderedType.v b/theories/Reals/ROrderedType.v
index 0a8d89c77..eeafbde9b 100644
--- a/theories/Reals/ROrderedType.v
+++ b/theories/Reals/ROrderedType.v
@@ -15,7 +15,7 @@ Local Open Scope R_scope.
 Lemma Req_dec : forall r1 r2:R, {r1 = r2} + {r1 <> r2}.
 Proof.
   intros; generalize (total_order_T r1 r2) Rlt_dichotomy_converse;
-    intuition eauto 3.
+    intuition eauto.
 Qed.
 
 Definition Reqb r1 r2 := if Req_dec r1 r2 then true else false.
diff --git a/theories/Structures/OrderedType.v b/theories/Structures/OrderedType.v
index f84cdf32c..beb10a833 100644
--- a/theories/Structures/OrderedType.v
+++ b/theories/Structures/OrderedType.v
@@ -223,7 +223,7 @@ Lemma Inf_lt : forall l x y, lt x y -> Inf y l -> Inf x l.
 Proof. exact (InfA_ltA lt_strorder). Qed.
 
 Lemma Inf_eq : forall l x y, eq x y -> Inf y l -> Inf x l.
-Proof. exact (InfA_eqA eq_equiv lt_strorder lt_compat). Qed.
+Proof. exact (InfA_eqA eq_equiv lt_compat). Qed.
 
 Lemma Sort_Inf_In : forall l x a, Sort l -> Inf a l -> In x l -> lt a x.
 Proof. exact (SortA_InfA_InA eq_equiv lt_strorder lt_compat). Qed.
@@ -396,7 +396,7 @@ Module KeyOrderedType(O:OrderedType).
   Qed.
 
   Lemma Inf_eq : forall l x x', eqk x x' -> Inf x' l -> Inf x l.
-  Proof. exact (InfA_eqA eqk_equiv ltk_strorder ltk_compat). Qed.
+  Proof. exact (InfA_eqA eqk_equiv ltk_compat). Qed.
 
   Lemma Inf_lt : forall l x x', ltk x x' -> Inf x' l -> Inf x l.
   Proof. exact (InfA_ltA ltk_strorder). Qed.
diff --git a/theories/Structures/OrdersLists.v b/theories/Structures/OrdersLists.v
index f83b63779..059992f5b 100644
--- a/theories/Structures/OrdersLists.v
+++ b/theories/Structures/OrdersLists.v
@@ -32,7 +32,7 @@ Lemma Inf_lt : forall l x y, lt x y -> Inf y l -> Inf x l.
 Proof. exact (InfA_ltA lt_strorder). Qed.
 
 Lemma Inf_eq : forall l x y, eq x y -> Inf y l -> Inf x l.
-Proof. exact (InfA_eqA eq_equiv lt_strorder lt_compat). Qed.
+Proof. exact (InfA_eqA eq_equiv lt_compat). Qed.
 
 Lemma Sort_Inf_In : forall l x a, Sort l -> Inf a l -> In x l -> lt a x.
 Proof. exact (SortA_InfA_InA eq_equiv lt_strorder lt_compat). Qed.
diff --git a/theories/ZArith/Znumtheory.v b/theories/ZArith/Znumtheory.v
index 6eb1a7093..2d4bfb2e3 100644
--- a/theories/ZArith/Znumtheory.v
+++ b/theories/ZArith/Znumtheory.v
@@ -575,30 +575,29 @@ Lemma prime_divisors :
   forall p:Z,
     prime p -> forall a:Z, (a | p) -> a = -1 \/ a = 1 \/ a = p \/ a = - p.
 Proof.
-  simple induction 1; intros.
+  destruct 1; intros.
   assert
     (a = - p \/ - p < a < -1 \/ a = -1 \/ a = 0 \/ a = 1 \/ 1 < a < p \/ a = p).
-  assert (Zabs a <= Zabs p). apply Zdivide_bounds; [ assumption | omega ].
-  generalize H3.
-  pattern (Zabs a) in |- *; apply Zabs_ind; pattern (Zabs p) in |- *;
-    apply Zabs_ind; intros; omega.
+  { assert (Zabs a <= Zabs p) as H2.
+      apply Zdivide_bounds; [ assumption | omega ].
+    revert H2.
+    pattern (Zabs a); apply Zabs_ind; pattern (Zabs p); apply Zabs_ind;
+    intros; omega. }
   intuition idtac.
   (* -p < a < -1 *)
-  absurd (rel_prime (- a) p); intuition.
-  inversion H3.
-  assert (- a | - a); auto with zarith.
-  assert (- a | p); auto with zarith.
-  generalize (H8 (- a) H9 H10); intuition idtac.
-  generalize (Zdivide_1 (- a) H11); intuition.
+  - absurd (rel_prime (- a) p); intuition.
+    inversion H2.
+    assert (- a | - a) by auto with zarith.
+    assert (- a | p) by auto with zarith.
+    apply H7, Zdivide_1 in H8; intuition.
   (* a = 0 *)
-  inversion H2. subst a; omega.
+  - inversion H1. subst a; omega.
   (* 1 < a < p *)
-  absurd (rel_prime a p); intuition.
-  inversion H3.
-  assert (a | a); auto with zarith.
-  assert (a | p); auto with zarith.
-  generalize (H8 a H9 H10); intuition idtac.
-  generalize (Zdivide_1 a H11); intuition.
+  - absurd (rel_prime a p); intuition.
+    inversion H2.
+    assert (a | a) by auto with zarith.
+    assert (a | p) by auto with zarith.
+    apply H7, Zdivide_1 in H8; intuition.
 Qed.
 
 (** A prime number is relatively prime with any number it does not divide *)
@@ -606,11 +605,10 @@ Qed.
 Lemma prime_rel_prime :
   forall p:Z, prime p -> forall a:Z, ~ (p | a) -> rel_prime p a.
 Proof.
-  simple induction 1; intros.
-  constructor; intuition.
-  elim (prime_divisors p H x H3); intuition; subst; auto with zarith.
-  absurd (p | a); auto with zarith.
-  absurd (p | a); intuition.
+  intros; constructor; intros; auto with zarith.
+  apply prime_divisors in H1; intuition; subst; auto with zarith.
+  - absurd (p | a); auto with zarith.
+  - absurd (p | a); intuition.
 Qed.
 
 Hint Resolve prime_rel_prime: zarith.
@@ -635,7 +633,7 @@ Lemma prime_mult :
   forall p:Z, prime p -> forall a b:Z, (p | a * b) -> (p | a) \/ (p | b).
 Proof.
   intro p; simple induction 1; intros.
-  case (Zdivide_dec p a); intuition.
+  case (Zdivide_dec p a); nintuition.
   right; apply Gauss with a; auto with zarith.
 Qed.