(************************************************************************)
(*  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        *)
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(* Certification of Imperative Programs / Jean-Christophe Filliātre *)

(* $Id: Tuples.v,v 1.1.2.1 2004/07/16 19:30:16 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 : (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 : (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 : (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 : (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 : (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 : (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 : (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 : (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 : (x1:T1)(x2:T2)(x3:T3)(P x1 x2 x3)
      	       	   -> (dep_tuple_3 T1 T2 T3 P).