Section PACO7.
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.
(** ** Signatures *)
Record sig7T  :=
exist7T {
proj7T0: @T0;
proj7T1: @T1 proj7T0;
proj7T2: @T2 proj7T0 proj7T1;
proj7T3: @T3 proj7T0 proj7T1 proj7T2;
proj7T4: @T4 proj7T0 proj7T1 proj7T2 proj7T3;
proj7T5: @T5 proj7T0 proj7T1 proj7T2 proj7T3 proj7T4;
proj7T6: @T6 proj7T0 proj7T1 proj7T2 proj7T3 proj7T4 proj7T5;
}.
Definition uncurry7  (R: rel7 T0 T1 T2 T3 T4 T5 T6): rel1 sig7T :=
fun x => R (proj7T0 x) (proj7T1 x) (proj7T2 x) (proj7T3 x) (proj7T4 x) (proj7T5 x) (proj7T6 x).
Definition curry7  (R: rel1 sig7T): rel7 T0 T1 T2 T3 T4 T5 T6 :=
fun x0 x1 x2 x3 x4 x5 x6 => R (@exist7T x0 x1 x2 x3 x4 x5 x6).
(** ** Predicates of Arity 7 *)
Definition paco7(gf : rel7 T0 T1 T2 T3 T4 T5 T6 -> rel7 T0 T1 T2 T3 T4 T5 T6)(r: rel7 T0 T1 T2 T3 T4 T5 T6) : rel7 T0 T1 T2 T3 T4 T5 T6 :=
curry7 (paco (fun R0 => uncurry7 (gf (curry7 R0))) (uncurry7 r)).
Definition upaco7(gf : rel7 T0 T1 T2 T3 T4 T5 T6 -> rel7 T0 T1 T2 T3 T4 T5 T6)(r: rel7 T0 T1 T2 T3 T4 T5 T6) := paco7 gf r \7/ r.
Arguments paco7 : clear implicits.
Arguments upaco7 : clear implicits.
Hint Unfold upaco7 : core.
Definition monotone7 (gf: rel7 T0 T1 T2 T3 T4 T5 T6 -> rel7 T0 T1 T2 T3 T4 T5 T6) :=
forall x0 x1 x2 x3 x4 x5 x6 r r' (IN: gf r x0 x1 x2 x3 x4 x5 x6) (LE: r <7= r'), gf r' x0 x1 x2 x3 x4 x5 x6.
Definition _monotone7 (gf: rel7 T0 T1 T2 T3 T4 T5 T6 -> rel7 T0 T1 T2 T3 T4 T5 T6) :=
forall r r'(LE: r <7= r'), gf r <7== gf r'.

Lemma monotone7_union (gf gf': rel7 T0 T1 T2 T3 T4 T5 T6 -> rel7 T0 T1 T2 T3 T4 T5 T6)
(MON1: monotone7 gf)
(MON2: monotone7 gf'):
monotone7 (gf \8/ gf').