Section PACO4.
Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Variable T2 : forall (x0: @T0) (x1: @T1 x0), Type.
Variable T3 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1), Type.
(** ** Signatures *)
Record sig4T  :=
exist4T {
proj4T0: @T0;
proj4T1: @T1 proj4T0;
proj4T2: @T2 proj4T0 proj4T1;
proj4T3: @T3 proj4T0 proj4T1 proj4T2;
}.
Definition uncurry4  (R: rel4 T0 T1 T2 T3): rel1 sig4T :=
fun x => R (proj4T0 x) (proj4T1 x) (proj4T2 x) (proj4T3 x).
Definition curry4  (R: rel1 sig4T): rel4 T0 T1 T2 T3 :=
fun x0 x1 x2 x3 => R (@exist4T x0 x1 x2 x3).

Lemma uncurry_bij1_4 r : curry4 (uncurry4 r) <4== r.