Section PACO13.
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.
Variable T9 : 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) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7), Type.
Variable T10 : 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) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8), Type.
Variable T11 : 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) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9), Type.
Variable T12 : 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) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9) (x11: @T11 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10), Type.
(** ** Signatures *)
Record sig13T  :=
exist13T {
proj13T0: @T0;
proj13T1: @T1 proj13T0;
proj13T2: @T2 proj13T0 proj13T1;
proj13T3: @T3 proj13T0 proj13T1 proj13T2;
proj13T4: @T4 proj13T0 proj13T1 proj13T2 proj13T3;
proj13T5: @T5 proj13T0 proj13T1 proj13T2 proj13T3 proj13T4;
proj13T6: @T6 proj13T0 proj13T1 proj13T2 proj13T3 proj13T4 proj13T5;
proj13T7: @T7 proj13T0 proj13T1 proj13T2 proj13T3 proj13T4 proj13T5 proj13T6;
proj13T8: @T8 proj13T0 proj13T1 proj13T2 proj13T3 proj13T4 proj13T5 proj13T6 proj13T7;
proj13T9: @T9 proj13T0 proj13T1 proj13T2 proj13T3 proj13T4 proj13T5 proj13T6 proj13T7 proj13T8;
proj13T10: @T10 proj13T0 proj13T1 proj13T2 proj13T3 proj13T4 proj13T5 proj13T6 proj13T7 proj13T8 proj13T9;
proj13T11: @T11 proj13T0 proj13T1 proj13T2 proj13T3 proj13T4 proj13T5 proj13T6 proj13T7 proj13T8 proj13T9 proj13T10;
proj13T12: @T12 proj13T0 proj13T1 proj13T2 proj13T3 proj13T4 proj13T5 proj13T6 proj13T7 proj13T8 proj13T9 proj13T10 proj13T11;
}.
Definition uncurry13  (R: rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12): rel1 sig13T :=
fun x => R (proj13T0 x) (proj13T1 x) (proj13T2 x) (proj13T3 x) (proj13T4 x) (proj13T5 x) (proj13T6 x) (proj13T7 x) (proj13T8 x) (proj13T9 x) (proj13T10 x) (proj13T11 x) (proj13T12 x).
Definition curry13  (R: rel1 sig13T): rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 :=
fun x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 => R (@exist13T x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12).
(** ** Predicates of Arity 13 *)
Definition paco13(gf : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 -> rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12)(r: rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 :=
curry13 (paco (fun R0 => uncurry13 (gf (curry13 R0))) (uncurry13 r)).
Definition upaco13(gf : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 -> rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12)(r: rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) := paco13 gf r \13/ r.
Arguments paco13 : clear implicits.
Arguments upaco13 : clear implicits.
Hint Unfold upaco13 : core.
Definition monotone13 (gf: rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 -> rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) :=
forall x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 r r' (IN: gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) (LE: r <13= r'), gf r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12.
Definition _monotone13 (gf: rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 -> rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) :=
forall r r'(LE: r <13= r'), gf r <13== gf r'.

Lemma upaco13_mon_bot (gf gf': rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 -> rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12
(REL: upaco13 gf bot13 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)
(LEgf: gf <14= gf'):
upaco13 gf' r' x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12.