Section PACO9.
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.
Variable T8 : 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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6), Type.
(** ** Signatures *)
Record sig9T  :=
exist9T {
proj9T0: @T0;
proj9T1: @T1 proj9T0;
proj9T2: @T2 proj9T0 proj9T1;
proj9T3: @T3 proj9T0 proj9T1 proj9T2;
proj9T4: @T4 proj9T0 proj9T1 proj9T2 proj9T3;
proj9T5: @T5 proj9T0 proj9T1 proj9T2 proj9T3 proj9T4;
proj9T6: @T6 proj9T0 proj9T1 proj9T2 proj9T3 proj9T4 proj9T5;
proj9T7: @T7 proj9T0 proj9T1 proj9T2 proj9T3 proj9T4 proj9T5 proj9T6;
proj9T8: @T8 proj9T0 proj9T1 proj9T2 proj9T3 proj9T4 proj9T5 proj9T6 proj9T7;
}.
Definition uncurry9  (R: rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8): rel1 sig9T :=
fun x => R (proj9T0 x) (proj9T1 x) (proj9T2 x) (proj9T3 x) (proj9T4 x) (proj9T5 x) (proj9T6 x) (proj9T7 x) (proj9T8 x).
Definition curry9  (R: rel1 sig9T): rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8 :=
fun x0 x1 x2 x3 x4 x5 x6 x7 x8 => R (@exist9T x0 x1 x2 x3 x4 x5 x6 x7 x8).
(** ** Predicates of Arity 9 *)
Definition paco9(gf : rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8 -> rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8)(r: rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8) : rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8 :=
curry9 (paco (fun R0 => uncurry9 (gf (curry9 R0))) (uncurry9 r)).
Definition upaco9(gf : rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8 -> rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8)(r: rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8) := paco9 gf r \9/ r.
Arguments paco9 : clear implicits.
Arguments upaco9 : clear implicits.
Hint Unfold upaco9 : core.
Definition monotone9 (gf: rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8 -> rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8) :=
forall x0 x1 x2 x3 x4 x5 x6 x7 x8 r r' (IN: gf r x0 x1 x2 x3 x4 x5 x6 x7 x8) (LE: r <9= r'), gf r' x0 x1 x2 x3 x4 x5 x6 x7 x8.
Definition _monotone9 (gf: rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8 -> rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8) :=
forall r r'(LE: r <9= r'), gf r <9== gf r'.

Lemma monotone9_union (gf gf': rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8 -> rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8)
(MON1: monotone9 gf)
(MON2: monotone9 gf'):
monotone9 (gf \10/ gf').