Section PACO6.
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.
(** ** Signatures *)
Record sig6T  :=
exist6T {
proj6T0: @T0;
proj6T1: @T1 proj6T0;
proj6T2: @T2 proj6T0 proj6T1;
proj6T3: @T3 proj6T0 proj6T1 proj6T2;
proj6T4: @T4 proj6T0 proj6T1 proj6T2 proj6T3;
proj6T5: @T5 proj6T0 proj6T1 proj6T2 proj6T3 proj6T4;
}.
Definition uncurry6  (R: rel6 T0 T1 T2 T3 T4 T5): rel1 sig6T :=
fun x => R (proj6T0 x) (proj6T1 x) (proj6T2 x) (proj6T3 x) (proj6T4 x) (proj6T5 x).
Definition curry6  (R: rel1 sig6T): rel6 T0 T1 T2 T3 T4 T5 :=
fun x0 x1 x2 x3 x4 x5 => R (@exist6T x0 x1 x2 x3 x4 x5).
(** ** Predicates of Arity 6 *)
Definition paco6(gf : rel6 T0 T1 T2 T3 T4 T5 -> rel6 T0 T1 T2 T3 T4 T5)(r: rel6 T0 T1 T2 T3 T4 T5) : rel6 T0 T1 T2 T3 T4 T5 :=
curry6 (paco (fun R0 => uncurry6 (gf (curry6 R0))) (uncurry6 r)).
Definition upaco6(gf : rel6 T0 T1 T2 T3 T4 T5 -> rel6 T0 T1 T2 T3 T4 T5)(r: rel6 T0 T1 T2 T3 T4 T5) := paco6 gf r \6/ r.
Arguments paco6 : clear implicits.
Arguments upaco6 : clear implicits.
Hint Unfold upaco6 : core.
Definition monotone6 (gf: rel6 T0 T1 T2 T3 T4 T5 -> rel6 T0 T1 T2 T3 T4 T5) :=
forall x0 x1 x2 x3 x4 x5 r r' (IN: gf r x0 x1 x2 x3 x4 x5) (LE: r <6= r'), gf r' x0 x1 x2 x3 x4 x5.
Definition _monotone6 (gf: rel6 T0 T1 T2 T3 T4 T5 -> rel6 T0 T1 T2 T3 T4 T5) :=
forall r r'(LE: r <6= r'), gf r <6== gf r'.

Lemma upaco6_mon_bot (gf gf': rel6 T0 T1 T2 T3 T4 T5 -> rel6 T0 T1 T2 T3 T4 T5) r' x0 x1 x2 x3 x4 x5
(REL: upaco6 gf bot6 x0 x1 x2 x3 x4 x5)
(LEgf: gf <7= gf'):
upaco6 gf' r' x0 x1 x2 x3 x4 x5.