Input TRS:
    1: app(nil(),k) -> k
    2: app(l,nil()) -> l
    3: app(cons(x,l),k) -> cons(x,app(l,k))
    4: sum(cons(x,nil())) -> cons(x,nil())
    5: sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
    6: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
    7: plus(0(),y) -> y
    8: plus(s(x),y) -> s(plus(x,y))
    9: sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
    10: pred(cons(s(x),nil())) -> cons(x,nil())
Number of strict rules: 10
Direct Order(PosReal,>,Poly) ... removes: 1 3 5 10 7 9 2
     s(x1)	weight: (/ 1 8) + x1
  pred(x1)	weight: x1
   sum(x1)	weight: x1
     0()	weight: 0
   nil()	weight: 0
  plus(x1,x2)	weight: (/ 1 4) + x1 + x2
  cons(x1,x2)	weight: (/ 71563 8) + x1 + x2
   app(x1,x2)	weight: (/ 127369 8) + 2 * x1 + x2
Number of strict rules: 3
Direct Order(PosReal,>,Poly) ... removes: 8
     s(x1)	weight: (/ 1 8) + x1
  pred(x1)	weight: x1
   sum(x1)	weight: x1
     0()	weight: 0
   nil()	weight: 0
  plus(x1,x2)	weight: (/ 4409 4) + 2 * x1 + x2
  cons(x1,x2)	weight: (/ 8817 8) + x1 + x2
   app(x1,x2)	weight: (/ 127369 8) + 2 * x1 + x2
Number of strict rules: 2
Direct Order(PosReal,>,Poly) ... failed.
Freezing sum
4: sum❆1_cons(x,nil()) -> cons(x,nil())
6: sum❆1_app(l,cons(x,cons(y,k))) -> sum❆1_app(l,sum❆1_cons(x,cons(y,k)))
11: sum(cons(_1,_2)) ->= sum❆1_cons(_1,_2)
12: sum(app(_1,_2)) ->= sum❆1_app(_1,_2)
Number of strict rules: 2
Direct Order(PosReal,>,Poly) ... removes: 12 11
sum❆1_app(x1,x2)	weight: (/ 33461 2) + x1 + x2
     s(x1)	weight: x1
sum❆1_cons(x1,x2)	weight: (/ 64571 2) + x1 + x2
  pred(x1)	weight: x1
   sum(x1)	weight: 2 * x1
     0()	weight: 0
   nil()	weight: 0
  plus(x1,x2)	weight: x1 + x2
  cons(x1,x2)	weight: (/ 64571 2) + x1 + x2
   app(x1,x2)	weight: (/ 16731 2) + x1 + x2
Number of strict rules: 2
Direct Order(PosReal,>,Poly) ... removes: 4 6
sum❆1_app(x1,x2)	weight: (/ 267681 16) + x1 + x2
     s(x1)	weight: x1
sum❆1_cons(x1,x2)	weight: (/ 3 16) + x1 + x2
  pred(x1)	weight: x1
   sum(x1)	weight: 2 * x1
     0()	weight: 0
   nil()	weight: 0
  plus(x1,x2)	weight: x1 + x2
  cons(x1,x2)	weight: (/ 1 8) + x1 + 2 * x2
   app(x1,x2)	weight: (/ 133841 16) + x1 + x2
Number of strict rules: 0
YES