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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(* Petit bench vite fait, mal fait *)
+
+Require Refine.
+
+
+(************************************************************************)
+
+Lemma essai : (x:nat)x=x.
+
+Refine (([x0:nat]Cases x0 of
+    O => ?
+  | (S p) => ?
+  end) :: (x:nat)x=x).  (* x0=x0 et x0=x0 *)
+
+Restart.
+
+Refine [x0:nat]<[n:nat]n=n>Case x0 of ? [p:nat]? end. (* OK *)
+
+Restart.
+
+Refine [x0:nat]<[n:nat]n=n>Cases x0 of O => ? | (S p) => ? end. (* OK *)
+
+Restart.
+
+(**
+Refine [x0:nat]Cases x0 of O => ? | (S p) => ? end. (* cannot be executed *)
+**)
+
+Abort.
+
+
+(************************************************************************)
+
+Lemma T : nat. 
+
+Refine (S ?).
+
+Abort.
+
+
+(************************************************************************)
+
+Lemma essai2 : (x:nat)x=x.
+
+Refine Fix f{f/1 : (x:nat)x=x := [x:nat]? }.
+
+Restart.
+
+Refine Fix f{f/1 : (x:nat)x=x :=
+  [x:nat]<[n:nat](eq nat n n)>Case x of ? [p:nat]? end}.
+
+Restart.
+
+Refine Fix f{f/1 : (x:nat)x=x :=
+  [x:nat]<[n:nat]n=n>Cases x of O => ? | (S p) => ? end}.
+
+Restart.
+
+Refine Fix f{f/1 : (x:nat)x=x :=
+  [x:nat]<[n:nat](eq nat n n)>Case x of
+       ?
+       [p:nat](f_equal nat nat S p p ?) end}.
+
+Restart.
+
+Refine Fix f{f/1 : (x:nat)x=x :=
+  [x:nat]<[n:nat](eq nat n n)>Cases x of
+       O => ?
+     | (S p) =>(f_equal nat nat S p p ?) end}.
+
+Abort.
+
+
+(************************************************************************)
+
+Lemma essai : nat.
+
+Parameter f : nat*nat -> nat -> nat. 
+
+Refine (f ? ([x:nat](? :: nat) O)).
+
+Restart.
+
+Refine (f ? O).
+
+Abort.
+
+
+(************************************************************************)
+
+Parameter P : nat -> Prop.
+
+Lemma essai : { x:nat | x=(S O) }.
+
+Refine (exist nat ? (S O) ?).  (* ECHEC *)
+
+Restart.
+
+(* mais si on contraint par le but alors ca marche : *)
+(* Remarque : on peut toujours faire ça *)
+Refine ((exist nat ? (S O) ?) :: { x:nat | x=(S O) }).
+
+Restart.
+
+Refine (exist nat [x:nat](x=(S O)) (S O) ?).
+
+Abort.
+
+
+(************************************************************************)
+
+Lemma essai : (n:nat){ x:nat | x=(S n) }.
+
+Refine [n:nat]<[n:nat]{x:nat|x=(S n)}>Case n of ? [p:nat]? end.
+
+Restart.
+
+Refine (([n:nat]Case n of ? [p:nat]? end) :: (n:nat){ x:nat | x=(S n) }).
+
+Restart.
+
+Refine [n:nat]<[n:nat]{x:nat|x=(S n)}>Cases n of O => ? | (S p) => ? end.
+
+Restart.
+
+Refine Fix f{f/1 :(n:nat){x:nat|x=(S n)} :=
+        [n:nat]<[n:nat]{x:nat|x=(S n)}>Case n of ? [p:nat]? end}.
+
+Restart.
+
+Refine Fix f{f/1 :(n:nat){x:nat|x=(S n)} :=
+        [n:nat]<[n:nat]{x:nat|x=(S n)}>Cases n of O => ? | (S p) => ? end}.
+
+Exists (S O). Trivial. 
+Elim (f0 p).
+Refine [x:nat][h:x=(S p)](exist nat [x:nat]x=(S (S p)) (S x) ?). 
+Rewrite h. Auto.
+Save.
+
+
+
+(* Quelques essais de recurrence bien fondée *)
+
+Require Wf.
+Require Wf_nat.
+
+Lemma essai_wf : nat->nat.
+
+Refine [x:nat](well_founded_induction
+      	       nat
+      	       lt ?
+               [_:nat]nat->nat
+	       [phi0:nat][w:(phi:nat)(lt phi phi0)->nat->nat](w x ?)
+	       x x).
+Exact lt_wf.
+
+Abort.
+
+
+Require Compare_dec.
+Require Lt.
+
+Lemma fibo : nat -> nat.
+Refine (well_founded_induction
+       nat
+       lt ?
+       [_:nat]nat
+       [x0:nat][fib:(x:nat)(lt x x0)->nat]
+         Cases (zerop x0) of 
+      	   (left _)   => (S O)
+      	 | (right h1) => Cases (zerop (pred x0)) of
+      	                   (left _)   => (S O)
+                         | (right h2) => (plus (fib (pred x0) ?)
+      	       	       	       	       	       (fib (pred (pred x0)) ?))
+		     end
+	 end).
+(*********
+Refine (well_founded_induction
+       nat
+       lt ?
+       [_:nat]nat
+       [x0:nat][fib:(x:nat)(lt x x0)->nat]
+         Cases x0 of 
+      	   O     => (S O)
+      	 | (S O) => (S O)
+      	 | (S (S p)) => (plus (fib (pred x0) ?)
+      	       	       	      (fib (pred (pred x0)) ?))
+	 end).
+***********)
+Exact lt_wf.
+Auto.
+Apply lt_trans with m:=(pred x0); Auto.
+Save.
+
+