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+(* Check forward dependencies *)
+
+Check [P:nat->Prop][Q][A:(P O)->Q][B:(n:nat)(P (S n))->Q][x]
+  <[_]Q>Cases x of 
+  | (exist O H) => (A H)
+  | (exist (S n) H) => (B n H)
+  end.
+
+(* Check dependencies in anonymous arguments (from FTA/listn.v) *)
+
+Inductive listn [A:Set] : nat->Set :=
+  niln: (listn A O)
+| consn: (a:A)(n:nat)(listn A n)->(listn A (S n)).
+
+Section Folding.
+Variables B, C : Set.
+Variable g : B -> C -> C.
+Variable c : C.
+
+Fixpoint foldrn [n:nat; bs:(listn B n)] : C :=
+  Cases bs of niln => c
+            | (consn b _ tl) => (g b (foldrn ? tl))
+  end.
+End Folding.
+
+(* -------------------------------------------------------------------- *)
+(*   Example to test patterns matching on dependent families            *)
+(* This exemple extracted from the developement done by Nacira Chabane  *)
+(* (equipe Paris 6)                                                     *)
+(* -------------------------------------------------------------------- *)
+
+
+Require Prelude.
+Require Logic_Type.
+
+Section Orderings.
+   Variable U: Type.
+   
+   Definition Relation :=  U -> U -> Prop.
+
+   Variable R: Relation.
+   
+   Definition Reflexive : Prop :=  (x: U) (R x x).
+   
+   Definition Transitive : Prop :=  (x,y,z: U) (R x y) -> (R y z) -> (R x z).
+   
+   Definition Symmetric : Prop :=  (x,y: U) (R x y) -> (R y x).
+   
+   Definition Antisymmetric : Prop :=
+      (x,y: U) (R x y) -> (R y x) -> x==y.
+   
+   Definition contains : Relation -> Relation -> Prop :=
+      [R,R': Relation] (x,y: U) (R' x y) -> (R x y).
+  Definition same_relation : Relation -> Relation -> Prop :=
+      [R,R': Relation] (contains R R') /\ (contains R' R).
+Inductive Equivalence : Prop :=
+         Build_Equivalence:
+           Reflexive -> Transitive -> Symmetric -> Equivalence.
+   
+   Inductive PER : Prop :=
+         Build_PER: Symmetric -> Transitive -> PER.
+   
+End Orderings.
+
+(***** Setoid  *******)
+
+Inductive Setoid : Type 
+  := Build_Setoid : (S:Type)(R:(Relation S))(Equivalence ? R) -> Setoid.
+
+Definition elem := [A:Setoid] let (S,R,e)=A in S.
+
+Grammar constr constr1 :=
+ elem  [ "|" constr0($s) "|"] -> [ (elem $s) ].
+
+Definition equal := [A:Setoid]
+   <[s:Setoid](Relation |s|)>let (S,R,e)=A in R.
+
+Grammar constr constr1 :=
+ equal   [ constr0($c) "=" "%" "S" constr0($c2) ] -> 
+         [ (equal ? $c $c2) ].
+
+
+Axiom prf_equiv : (A:Setoid)(Equivalence |A| (equal A)).
+Axiom prf_refl : (A:Setoid)(Reflexive |A| (equal A)).
+Axiom prf_sym : (A:Setoid)(Symmetric |A| (equal A)).
+Axiom prf_trans : (A:Setoid)(Transitive |A| (equal A)).
+
+Section Maps.
+Variables A,B: Setoid.
+
+Definition Map_law := [f:|A| -> |B|]
+    (x,y:|A|) x =%S y -> (f x) =%S (f y).
+
+Inductive Map : Type :=
+    Build_Map : (f:|A| -> |B|)(p:(Map_law f))Map.
+
+Definition explicit_ap := [m:Map] <|A| -> |B|>Match m with 
+         [f:?][p:?]f end.
+
+Axiom pres : (m:Map)(Map_law (explicit_ap m)).
+
+Definition ext := [f,g:Map]
+         (x:|A|) (explicit_ap f x) =%S (explicit_ap g x).
+
+Axiom Equiv_map_eq : (Equivalence Map ext).
+
+Definition Map_setoid := (Build_Setoid Map ext Equiv_map_eq).
+
+End Maps.
+
+Notation ap := (explicit_ap ? ?). 
+
+Grammar constr constr8 :=
+  map_setoid [ constr7($c1) "=>" constr8($c2) ] 
+                 -> [ (Map_setoid $c1 $c2) ].
+
+
+Definition ap2 := [A,B,C:Setoid][f:|(A=>(B=>C))|][a:|A|] (ap (ap f a)).
+
+
+(*****    posint     ******)
+
+Inductive posint : Type
+        := Z : posint | Suc : posint -> posint.
+
+Axiom f_equal : (A,B:Type)(f:A->B)(x,y:A) x==y -> (f x)==(f y).
+Axiom eq_Suc : (n,m:posint) n==m -> (Suc n)==(Suc m).
+
+(* The predecessor function *)
+
+Definition pred : posint->posint
+     := [n:posint](<posint>Case n of (* Z *) Z 
+                            (* Suc u *) [u:posint]u end).
+
+Axiom pred_Sucn : (m:posint) m==(pred (Suc m)).
+Axiom eq_add_Suc : (n,m:posint) (Suc n)==(Suc m) -> n==m.
+Axiom not_eq_Suc : (n,m:posint) ~(n==m) -> ~((Suc n)==(Suc m)).
+
+
+Definition IsSuc : posint->Prop
+  := [n:posint](<Prop>Case n of (* Z *) False
+                          (* Suc p *) [p:posint]True end).
+Definition IsZero :posint->Prop :=
+	[n:posint]<Prop>Match n with 
+		True
+		[p:posint][H:Prop]False end.
+
+Axiom Z_Suc : (n:posint) ~(Z==(Suc n)).
+Axiom  Suc_Z: (n:posint) ~(Suc n)==Z.
+Axiom n_Sucn : (n:posint) ~(n==(Suc n)).
+Axiom Sucn_n : (n:posint) ~(Suc n)==n.
+Axiom eqT_symt : (a,b:posint) ~(a==b)->~(b==a).
+
+
+(*******  Dsetoid *****)
+
+Definition Decidable :=[A:Type][R:(Relation A)]
+		(x,y:A)(R x y) \/ ~(R x y).
+
+
+Record DSetoid : Type :=
+{Set_of : Setoid;
+ prf_decid : (Decidable |Set_of| (equal Set_of))}.
+
+(* example de Dsetoide d'entiers *)
+
+
+Axiom eqT_equiv : (Equivalence posint (eqT posint)).
+Axiom Eq_posint_deci : (Decidable posint (eqT posint)).
+
+(* Dsetoide des posint*)
+
+Definition Set_of_posint := (Build_Setoid posint (eqT posint) eqT_equiv).
+
+Definition Dposint := (Build_DSetoid Set_of_posint Eq_posint_deci).
+
+
+
+(**************************************)
+
+
+(* Definition des signatures *)
+(* une signature est un ensemble d'operateurs muni
+ de l'arite de chaque operateur *)
+
+
+Section Sig.
+
+Record Signature :Type :=
+{Sigma : DSetoid; 
+ Arity : (Map (Set_of Sigma) (Set_of Dposint))}.
+
+Variable S:Signature.
+
+
+
+Variable Var : DSetoid.
+
+Mutual Inductive TERM : Type :=
+	  var : |(Set_of Var)| -> TERM
+	| oper : (op: |(Set_of (Sigma S))| ) (LTERM (ap (Arity S) op)) -> TERM
+with
+	LTERM : posint -> Type :=
+	  nil : (LTERM Z)
+	| cons : TERM -> (n:posint)(LTERM n) -> (LTERM (Suc n)).
+
+
+
+(* -------------------------------------------------------------------- *)
+(*                  Examples                                            *)
+(* -------------------------------------------------------------------- *)
+
+
+Parameter t1,t2: TERM.
+
+Type 
+ Cases t1 t2 of 
+ | (var  v1) (var v2) => True
+ | (oper op1 l1) (oper op2 l2) => False
+ | _ _ => False 
+ end.
+
+
+
+Parameter n2:posint.
+Parameter l1, l2:(LTERM n2).
+
+Type 
+ Cases l1 l2  of
+    nil            nil => True 
+ |  (cons v m y)   nil => False
+ |  _               _  => False
+end.
+
+
+Type Cases l1 l2 of
+    nil              nil      => True 
+ | (cons u n x)  (cons v m y) =>False
+ |   _               _        => False
+end.
+
+
+
+Definition equalT [t1:TERM]:TERM->Prop :=
+[t2:TERM] 
+ Cases t1 t2 of 
+   (var  v1)        (var v2)    => True
+ | (oper op1 l1)  (oper op2 l2) => False
+ |  _                   _       => False 
+ end.
+
+Definition EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
+[n2:posint][l2:(LTERM n2)]
+ Cases l1 l2 of
+    nil                 nil             => True 
+ | (cons t1 n1' l1')  (cons t2 n2' l2') => False
+ | _                    _               => False
+end.
+
+
+Reset equalT.
+(* ------------------------------------------------------------------*)
+(*          Initial exemple (without patterns)                       *)
+(*-------------------------------------------------------------------*)
+
+Fixpoint equalT [t1:TERM]:TERM->Prop :=
+<TERM->Prop>Case t1 of 
+	(*var*) [v1:|(Set_of Var)|][t2:TERM]
+		<Prop>Case t2 of
+		(*var*)[v2:|(Set_of Var)|] (v1 =%S v2)
+		(*oper*)[op2:|(Set_of (Sigma S))|][_:(LTERM (ap (Arity S) op2))]False
+		end
+	(*oper*)[op1:|(Set_of (Sigma S))|]
+                [l1:(LTERM (ap (Arity S) op1))][t2:TERM]
+                <Prop>Case t2 of
+		(*var*)[v2:|(Set_of Var)|]False
+		(*oper*)[op2:|(Set_of (Sigma S))|]
+                        [l2:(LTERM (ap (Arity S) op2))]
+			((op1=%S op2)/\ (EqListT (ap (Arity S) op1) l1 (ap (Arity S) op2) l2))
+		end
+end
+with EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
+<[_:posint](n2:posint)(LTERM n2)->Prop>Case l1 of
+	(*nil*) [n2:posint][l2:(LTERM n2)]
+		<[_:posint]Prop>Case l2 of
+		(*nil*)True
+	 	(*cons*)[t2:TERM][n2':posint][l2':(LTERM n2')]False
+	end
+	(*cons*)[t1:TERM][n1':posint][l1':(LTERM n1')]
+		[n2:posint][l2:(LTERM n2)]
+		<[_:posint]Prop>Case l2 of
+		(*nil*) False
+		(*cons*)[t2:TERM][n2':posint][l2':(LTERM n2')]
+			((equalT t1 t2) /\ (EqListT n1' l1' n2' l2'))
+		end
+end.
+
+
+(* ---------------------------------------------------------------- *)
+(*                Version with simple patterns                      *)
+(* ---------------------------------------------------------------- *)
+Reset equalT.
+
+Fixpoint equalT [t1:TERM]:TERM->Prop :=
+Cases t1 of 
+  (var  v1) => [t2:TERM]
+	        Cases t2 of
+		     (var v2) => (v1 =%S v2)
+		 | (oper op2 _) =>False
+		 end
+| (oper op1 l1) => [t2:TERM]
+		    Cases t2 of
+	              (var _) => False
+                    | (oper op2 l2) => (op1=%S op2)
+                      /\ (EqListT (ap (Arity S) op1) l1 (ap (Arity S) op2) l2)
+		    end
+end
+with EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
+<[_:posint](n2:posint)(LTERM n2)->Prop>Cases l1 of
+  nil => [n2:posint][l2:(LTERM n2)]
+          Cases l2 of
+              nil => True
+	  | _  => False
+	  end
+| (cons t1 n1' l1') => [n2:posint][l2:(LTERM n2)]
+		        Cases l2 of
+		           nil =>False
+		        | (cons t2 n2' l2') => (equalT t1 t2) 
+                                                /\ (EqListT n1' l1' n2' l2')
+		end
+end.
+
+
+Reset equalT.
+
+Fixpoint equalT [t1:TERM]:TERM->Prop :=
+Cases t1 of 
+  (var  v1) => [t2:TERM]
+	        Cases t2 of
+		     (var v2) => (v1 =%S v2)
+		 | (oper op2 _) =>False
+		 end
+| (oper op1 l1) => [t2:TERM]
+		    Cases t2 of
+	              (var _) => False
+                    | (oper op2 l2) => (op1=%S op2)
+                      /\ (EqListT (ap (Arity S) op1) l1 (ap (Arity S) op2) l2)
+		    end
+end
+with EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
+[n2:posint][l2:(LTERM n2)]
+Cases l1 of
+  nil => 
+          Cases l2 of
+              nil => True
+	  | _  => False
+	  end
+| (cons t1 n1' l1') =>  Cases l2 of
+		           nil =>False
+		        | (cons t2 n2' l2') => (equalT t1 t2) 
+                                                /\ (EqListT n1' l1' n2' l2')
+		end
+end.
+
+(* ---------------------------------------------------------------- *)
+(*                  Version with multiple patterns                  *)
+(* ---------------------------------------------------------------- *)
+Reset equalT.
+
+Fixpoint equalT [t1:TERM]:TERM->Prop :=
+[t2:TERM] 
+ Cases t1 t2 of 
+   (var  v1)       (var v2)     => (v1 =%S v2)
+
+ | (oper op1 l1)  (oper op2 l2) => 
+     (op1=%S op2) /\ (EqListT (ap (Arity S) op1) l1 (ap (Arity S) op2) l2)
+
+ |  _               _ => False 
+ end
+
+with EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
+[n2:posint][l2:(LTERM n2)]
+ Cases l1 l2 of
+    nil                nil              => True 
+ | (cons t1 n1' l1')  (cons t2 n2' l2') => (equalT t1 t2) 
+                                          /\ (EqListT n1' l1' n2'  l2')
+ | _ _ => False
+end.
+
+
+(* ------------------------------------------------------------------ *)
+
+End Sig.
+
+(* Exemple soumis par Bruno *)
+
+Definition bProp [b:bool] : Prop :=
+  if b then True else False.
+
+Definition f0 [F:False;ty:bool]: (bProp ty) :=
+  <[_:bool][ty:bool](bProp ty)>Cases ty ty of
+    true true => I
+  | _ false => F
+  | _ true => I
+  end.