Definition tuple_1 := [X:Set]X.
Definition tuple_2 := prod.
Definition Build_tuple_2 := pair.
Implicits Build_tuple_2.
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 }.
Implicits Build_tuple_3.
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 }.
Implicits Build_tuple_4.
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 }.
Implicits Build_tuple_5.
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 }.
Implicits Build_tuple_6.
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 }.
Implicits Build_tuple_7.
Definition sig_1 := sig.
Definition exist_1 := exist.
Implicits exist_1 [1].
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).
Implicits exist_2 [1 2].
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).
Implicits exist_3 [1 2 3].
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).
Implicits exist_4 [1 2 3 4].
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).
Implicits exist_5 [1 2 3 4 5].
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).
Implicits exist_6 [1 2 3 4 5 6].
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).
Implicits exist_7 [1 2 3 4 5 6 7].
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).
Implicits exist_8 [1 2 3 4 5 6 7 8].