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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        *)
+(************************************************************************)
+
+(* Certification of Imperative Programs / Jean-Christophe Filliātre *)
+
+(* $Id: Tuples.v,v 1.2.2.1 2004/07/16 19:30:00 herbelin Exp $ *)
+
+(* Tuples *)
+
+Definition tuple_1 (X:Set) := X.
+Definition tuple_2 := prod.
+Definition Build_tuple_2 := pair.
+Definition proj_2_1 := fst.
+Definition proj_2_2 := snd.
+
+Record tuple_3 (T1 T2 T3:Set) : Set := 
+  {proj_3_1 : T1; proj_3_2 : T2; proj_3_3 : T3}.
+
+Record tuple_4 (T1 T2 T3 T4:Set) : Set := 
+  {proj_4_1 : T1; proj_4_2 : T2; proj_4_3 : T3; proj_4_4 : T4}.
+
+Record tuple_5 (T1 T2 T3 T4 T5:Set) : Set := 
+  {proj_5_1 : T1; proj_5_2 : T2; proj_5_3 : T3; proj_5_4 : T4; proj_5_5 : T5}.
+
+Record tuple_6 (T1 T2 T3 T4 T5 T6:Set) : Set := 
+  {proj_6_1 : T1;
+   proj_6_2 : T2;
+   proj_6_3 : T3;
+   proj_6_4 : T4;
+   proj_6_5 : T5;
+   proj_6_6 : T6}.
+
+Record tuple_7 (T1 T2 T3 T4 T5 T6 T7:Set) : Set := 
+  {proj_7_1 : T1;
+   proj_7_2 : T2;
+   proj_7_3 : T3;
+   proj_7_4 : T4;
+   proj_7_5 : T5;
+   proj_7_6 : T6;
+   proj_7_7 : T7}.
+
+
+(* Existentials *)
+
+Definition sig_1 := sig.
+Definition exist_1 := exist.
+
+Inductive sig_2 (T1 T2:Set) (P:T1 -> T2 -> Prop) : Set :=
+    exist_2 : forall (x1:T1) (x2:T2), P x1 x2 -> sig_2 T1 T2 P.
+
+Inductive sig_3 (T1 T2 T3:Set) (P:T1 -> T2 -> T3 -> Prop) : Set :=
+    exist_3 : forall (x1:T1) (x2:T2) (x3:T3), P x1 x2 x3 -> sig_3 T1 T2 T3 P.
+
+
+Inductive sig_4 (T1 T2 T3 T4:Set) (P:T1 -> T2 -> T3 -> T4 -> Prop) : Set :=
+    exist_4 :
+      forall (x1:T1) (x2:T2) (x3:T3) (x4:T4),
+        P x1 x2 x3 x4 -> sig_4 T1 T2 T3 T4 P.
+
+Inductive sig_5 (T1 T2 T3 T4 T5:Set) (P:T1 -> T2 -> T3 -> T4 -> T5 -> Prop) :
+Set :=
+    exist_5 :
+      forall (x1:T1) (x2:T2) (x3:T3) (x4:T4) (x5:T5),
+        P x1 x2 x3 x4 x5 -> sig_5 T1 T2 T3 T4 T5 P.
+
+Inductive sig_6 (T1 T2 T3 T4 T5 T6:Set)
+(P:T1 -> T2 -> T3 -> T4 -> T5 -> T6 -> Prop) : Set :=
+    exist_6 :
+      forall (x1:T1) (x2:T2) (x3:T3) (x4:T4) (x5:T5) 
+        (x6:T6), P x1 x2 x3 x4 x5 x6 -> sig_6 T1 T2 T3 T4 T5 T6 P.
+
+Inductive sig_7 (T1 T2 T3 T4 T5 T6 T7:Set)
+(P:T1 -> T2 -> T3 -> T4 -> T5 -> T6 -> T7 -> Prop) : Set :=
+    exist_7 :
+      forall (x1:T1) (x2:T2) (x3:T3) (x4:T4) (x5:T5) 
+        (x6:T6) (x7:T7),
+        P x1 x2 x3 x4 x5 x6 x7 -> sig_7 T1 T2 T3 T4 T5 T6 T7 P.
+
+Inductive sig_8 (T1 T2 T3 T4 T5 T6 T7 T8:Set)
+(P:T1 -> T2 -> T3 -> T4 -> T5 -> T6 -> T7 -> T8 -> Prop) : Set :=
+    exist_8 :
+      forall (x1:T1) (x2:T2) (x3:T3) (x4:T4) (x5:T5) 
+        (x6:T6) (x7:T7) (x8:T8),
+        P x1 x2 x3 x4 x5 x6 x7 x8 -> sig_8 T1 T2 T3 T4 T5 T6 T7 T8 P.      
+
+Inductive dep_tuple_2 (T1 T2:Set) (P:T1 -> T2 -> Set) : Set :=
+    Build_dep_tuple_2 :
+      forall (x1:T1) (x2:T2), P x1 x2 -> dep_tuple_2 T1 T2 P.
+
+Inductive dep_tuple_3 (T1 T2 T3:Set) (P:T1 -> T2 -> T3 -> Set) : Set :=
+    Build_dep_tuple_3 :
+      forall (x1:T1) (x2:T2) (x3:T3), P x1 x2 x3 -> dep_tuple_3 T1 T2 T3 P.
+