Input TRS:
    1: a__nats() -> cons(|0|(),incr(nats()))
    2: a__pairs() -> cons(|0|(),incr(odds()))
    3: a__odds() -> a__incr(a__pairs())
    4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
    5: a__head(cons(X,XS)) -> mark(X)
    6: a__tail(cons(X,XS)) -> mark(XS)
    7: mark(nats()) -> a__nats()
    8: mark(pairs()) -> a__pairs()
    9: mark(odds()) -> a__odds()
    10: mark(incr(X)) -> a__incr(mark(X))
    11: mark(head(X)) -> a__head(mark(X))
    12: mark(tail(X)) -> a__tail(mark(X))
    13: mark(|0|()) -> |0|()
    14: mark(s(X)) -> s(mark(X))
    15: mark(nil()) -> nil()
    16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    17: a__nats() -> nats()
    18: a__pairs() -> pairs()
    19: a__odds() -> odds()
    20: a__incr(X) -> incr(X)
    21: a__head(X) -> head(X)
    22: a__tail(X) -> tail(X)
Number of strict rules: 22
Direct Order(PosReal,>,Poly) ... removes: 5 6
   |0|()	weight: 0
  incr(x1)	weight: x1
     s(x1)	weight: x1
a__incr(x1)	weight: x1
 pairs()	weight: 0
a__nats()	weight: 0
a__odds()	weight: 0
  tail(x1)	weight: (/ 1 4) + x1
  odds()	weight: 0
   nil()	weight: 0
  mark(x1)	weight: x1
  nats()	weight: 0
  head(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: x1 + x2
a__pairs()	weight: 0
a__tail(x1)	weight: (/ 1 4) + x1
a__head(x1)	weight: (/ 1 4) + x1
Number of strict rules: 20
Direct Order(PosReal,>,Poly) ... removes: 21 22 12 11
   |0|()	weight: 0
  incr(x1)	weight: 2 * x1
     s(x1)	weight: x1
a__incr(x1)	weight: 2 * x1
 pairs()	weight: 0
a__nats()	weight: 0
a__odds()	weight: 0
  tail(x1)	weight: (/ 1 4) + x1
  odds()	weight: 0
   nil()	weight: 0
  mark(x1)	weight: 2 * x1
  nats()	weight: 0
  head(x1)	weight: (/ 117617 4) + x1
  cons(x1,x2)	weight: x1 + x2
a__pairs()	weight: 0
a__tail(x1)	weight: (/ 3 8) + x1
a__head(x1)	weight: (/ 470467 8) + x1
Number of strict rules: 16
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #mark(odds()) -> #a__odds()
   #2: #mark(s(X)) -> #mark(X)
   #3: #mark(nats()) -> #a__nats()
   #4: #mark(incr(X)) -> #a__incr(mark(X))
   #5: #mark(incr(X)) -> #mark(X)
   #6: #mark(cons(X1,X2)) -> #mark(X1)
   #7: #a__odds() -> #a__incr(a__pairs())
   #8: #a__odds() -> #a__pairs()
   #9: #mark(pairs()) -> #a__pairs()
   #10: #a__incr(cons(X,XS)) -> #mark(X)
Number of SCCs: 1, DPs: 7, edges: 23
	SCC { #1 #2 #4..7 #10 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
   |0|()	weight: (/ 1 8)
  incr(x1)	weight: x1
     s(x1)	weight: x1
a__incr(x1)	weight: x1
 pairs()	weight: (/ 1 2)
a__nats()	weight: (/ 1 2)
#a__pairs()	weight: 0
a__odds()	weight: (/ 1 2)
  tail(x1)	weight: 0
#mark(x1)	weight: (/ 1 4) + x1
  odds()	weight: (/ 1 2)
   nil()	weight: (/ 1 8)
#a__nats()	weight: 0
  mark(x1)	weight: x1
  nats()	weight: (/ 1 2)
#a__incr(x1)	weight: x1
  head(x1)	weight: 0
  cons(x1,x2)	weight: max{x2, (/ 3 8) + x1}
#a__odds()	weight: (/ 5 8)
a__pairs()	weight: (/ 1 2)
a__tail(x1)	weight: 0
a__head(x1)	weight: 0
    Usable rules: { 1..4 7..10 13..20 }
    Removed DPs: #1 #4 #6 #7 #10
Number of SCCs: 1, DPs: 2, edges: 4
	SCC { #2 #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
  incr(x1)	weight: (/ 3 4) + x1
     s(x1)	weight: (/ 1 4) + x1
a__incr(x1)	weight: (/ 1 2)
 pairs()	weight: 0
a__nats()	weight: 0
#a__pairs()	weight: 0
a__odds()	weight: 0
  tail(x1)	weight: 0
#mark(x1)	weight: x1
  odds()	weight: 0
   nil()	weight: 0
#a__nats()	weight: 0
  mark(x1)	weight: (/ 1 4)
  nats()	weight: 0
#a__incr(x1)	weight: 0
  head(x1)	weight: 0
  cons(x1,x2)	weight: (/ 3 4)
#a__odds()	weight: 0
a__pairs()	weight: 0
a__tail(x1)	weight: 0
a__head(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #2 #5
Number of SCCs: 0, DPs: 0, edges: 0
YES