Section PACO8.
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.
Variable T4 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2), Type.
Variable T5 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3), Type.
Variable T6 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4), Type.
Variable T7 : forall (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4) (x6: @T6 x0 x1 x2 x3 x4 x5), Type.
(** ** Signatures *)
Record sig8T  :=
exist8T {
proj8T0: @T0;
proj8T1: @T1 proj8T0;
proj8T2: @T2 proj8T0 proj8T1;
proj8T3: @T3 proj8T0 proj8T1 proj8T2;
proj8T4: @T4 proj8T0 proj8T1 proj8T2 proj8T3;
proj8T5: @T5 proj8T0 proj8T1 proj8T2 proj8T3 proj8T4;
proj8T6: @T6 proj8T0 proj8T1 proj8T2 proj8T3 proj8T4 proj8T5;
proj8T7: @T7 proj8T0 proj8T1 proj8T2 proj8T3 proj8T4 proj8T5 proj8T6;
}.
Definition uncurry8  (R: rel8 T0 T1 T2 T3 T4 T5 T6 T7): rel1 sig8T :=
fun x => R (proj8T0 x) (proj8T1 x) (proj8T2 x) (proj8T3 x) (proj8T4 x) (proj8T5 x) (proj8T6 x) (proj8T7 x).
Definition curry8  (R: rel1 sig8T): rel8 T0 T1 T2 T3 T4 T5 T6 T7 :=
fun x0 x1 x2 x3 x4 x5 x6 x7 => R (@exist8T x0 x1 x2 x3 x4 x5 x6 x7).
(** ** Predicates of Arity 8 *)
Definition paco8(gf : rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7)(r: rel8 T0 T1 T2 T3 T4 T5 T6 T7) : rel8 T0 T1 T2 T3 T4 T5 T6 T7 :=
curry8 (paco (fun R0 => uncurry8 (gf (curry8 R0))) (uncurry8 r)).
Definition upaco8(gf : rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7)(r: rel8 T0 T1 T2 T3 T4 T5 T6 T7) := paco8 gf r \8/ r.
Arguments paco8 : clear implicits.
Arguments upaco8 : clear implicits.
Hint Unfold upaco8 : core.
Definition monotone8 (gf: rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7) :=
forall x0 x1 x2 x3 x4 x5 x6 x7 r r' (IN: gf r x0 x1 x2 x3 x4 x5 x6 x7) (LE: r <8= r'), gf r' x0 x1 x2 x3 x4 x5 x6 x7.
Definition _monotone8 (gf: rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7) :=
forall r r'(LE: r <8= r'), gf r <8== gf r'.

Lemma monotone8_union (gf gf': rel8 T0 T1 T2 T3 T4 T5 T6 T7 -> rel8 T0 T1 T2 T3 T4 T5 T6 T7)
(MON1: monotone8 gf)
(MON2: monotone8 gf'):
monotone8 (gf \9/ gf').