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+
+(*************************************************************)
+(*      This file is distributed under the terms of the      *)
+(*      GNU Lesser General Public License Version 2.1        *)
+(*************************************************************)
+(*    Benjamin.Gregoire@inria.fr Laurent.Thery@inria.fr      *)
+(*************************************************************)
+
+
+(**********************************************************************
+       Tactic.v                                                                                    
+          Useful tactics                                                                       
+ **********************************************************************)
+
+(**************************************
+  A simple tactic to end a proof 
+**************************************)
+Ltac  finish := intros; auto; trivial; discriminate.
+
+
+(**************************************
+ A tactic for proof by contradiction
+     with contradict H 
+         H: ~A |-   B      gives           |-   A
+         H: ~A |- ~ B     gives  H: B |-   A
+         H:   A |-   B      gives           |- ~ A
+         H:   A |-   B      gives           |- ~ A
+         H:   A |- ~ B     gives  H: A |- ~ A
+**************************************)
+
+Ltac  contradict name := 
+     let term := type of name in (
+     match term with 
+       (~_) => 
+          match goal with 
+            |- ~ _  => let x := fresh in
+                     (intros x; case name; 
+                      generalize x; clear x name;
+                      intro name)
+          | |- _    => case name; clear name
+          end
+     | _ => 
+          match goal with 
+            |- ~ _  => let x := fresh in
+                    (intros x;  absurd term;
+                       [idtac | exact name]; generalize x; clear x name;
+                       intros name)
+          | |- _    => generalize name; absurd term;
+                       [idtac | exact name]; clear name
+          end
+     end).
+
+
+(**************************************
+  A tactic to do case analysis keeping the equality
+**************************************)
+
+Ltac case_eq name :=
+  generalize (refl_equal name); pattern name at -1 in |- *; case name.
+
+
+(**************************************
+ A tactic to use f_equal? theorems 
+**************************************)
+
+Ltac eq_tac := 
+ match goal with
+      |-  (?g _ = ?g _) => apply f_equal with (f := g)
+ |     |-  (?g ?X _ = ?g  ?X _) => apply f_equal with (f := g X)
+ |     |-  (?g _ _ = ?g  _ _) => apply f_equal2 with (f := g)
+ |     |-  (?g ?X ?Y _ = ?g ?X ?Y _) => apply f_equal with (f := g X Y)
+ |     |-  (?g ?X _ _ = ?g ?X _ _) => apply f_equal2 with (f := g X)
+ |     |-  (?g _ _ _ = ?g _ _ _) => apply f_equal3 with (f := g)
+ |     |-  (?g ?X ?Y ?Z _ = ?g ?X ?Y ?Z _) => apply f_equal with (f := g X Y Z)
+ |     |-  (?g ?X ?Y _ _ = ?g ?X ?Y _ _) => apply f_equal2 with (f := g X Y)
+ |     |-  (?g ?X _ _ _ = ?g ?X _ _ _) => apply f_equal3 with (f := g X)
+ |     |-  (?g _ _ _ _ _ = ?g _ _ _ _) => apply f_equal4 with (f := g)
+ end.
+
+(************************************** 
+ A stupid tactic that tries auto also after applying sym_equal
+**************************************)
+
+Ltac sauto := (intros; apply sym_equal; auto; fail) || auto.