Input TRS:
    1: a__fib(N) -> a__sel(mark(N),a__fib1(s(0()),s(0())))
    2: a__fib1(X,Y) -> cons(mark(X),fib1(Y,add(X,Y)))
    3: a__add(0(),X) -> mark(X)
    4: a__add(s(X),Y) -> s(a__add(mark(X),mark(Y)))
    5: a__sel(0(),cons(X,XS)) -> mark(X)
    6: a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS))
    7: mark(fib(X)) -> a__fib(mark(X))
    8: mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
    9: mark(fib1(X1,X2)) -> a__fib1(mark(X1),mark(X2))
    10: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
    11: mark(s(X)) -> s(mark(X))
    12: mark(0()) -> 0()
    13: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    14: a__fib(X) -> fib(X)
    15: a__sel(X1,X2) -> sel(X1,X2)
    16: a__fib1(X1,X2) -> fib1(X1,X2)
    17: a__add(X1,X2) -> add(X1,X2)
Number of strict rules: 17
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__fib1(X,Y) -> #mark(X)
   #2: #a__sel(s(N),cons(X,XS)) -> #a__sel(mark(N),mark(XS))
   #3: #a__sel(s(N),cons(X,XS)) -> #mark(N)
   #4: #a__sel(s(N),cons(X,XS)) -> #mark(XS)
   #5: #mark(cons(X1,X2)) -> #mark(X1)
   #6: #mark(fib1(X1,X2)) -> #a__fib1(mark(X1),mark(X2))
   #7: #mark(fib1(X1,X2)) -> #mark(X1)
   #8: #mark(fib1(X1,X2)) -> #mark(X2)
   #9: #mark(s(X)) -> #mark(X)
   #10: #mark(fib(X)) -> #a__fib(mark(X))
   #11: #mark(fib(X)) -> #mark(X)
   #12: #mark(add(X1,X2)) -> #a__add(mark(X1),mark(X2))
   #13: #mark(add(X1,X2)) -> #mark(X1)
   #14: #mark(add(X1,X2)) -> #mark(X2)
   #15: #a__sel(0(),cons(X,XS)) -> #mark(X)
   #16: #a__add(0(),X) -> #mark(X)
   #17: #a__fib(N) -> #a__sel(mark(N),a__fib1(s(0()),s(0())))
   #18: #a__fib(N) -> #mark(N)
   #19: #a__fib(N) -> #a__fib1(s(0()),s(0()))
   #20: #mark(sel(X1,X2)) -> #a__sel(mark(X1),mark(X2))
   #21: #mark(sel(X1,X2)) -> #mark(X1)
   #22: #mark(sel(X1,X2)) -> #mark(X2)
   #23: #a__add(s(X),Y) -> #a__add(mark(X),mark(Y))
   #24: #a__add(s(X),Y) -> #mark(X)
   #25: #a__add(s(X),Y) -> #mark(Y)
Number of SCCs: 1, DPs: 25, edges: 246
	SCC { #1..25 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     s(x1)	weight: x1
#a__add(x1,x2)	weight: max{x2, x1}
a__add(x1,x2)	weight: max{x2, x1}
#a__fib1(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}
  fib1(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
   fib(x1)	weight: (/ 5 8) + x1
#mark(x1)	weight: x1
     0()	weight: (/ 1 8)
   sel(x1,x2)	weight: max{(/ 1 8) + x2, x1}
#a__sel(x1,x2)	weight: max{x2, x1}
  mark(x1)	weight: x1
a__sel(x1,x2)	weight: max{(/ 1 8) + x2, x1}
#a__fib(x1)	weight: (/ 1 2) + x1
  cons(x1,x2)	weight: max{x2, (/ 1 8) + x1}
a__fib(x1)	weight: (/ 5 8) + x1
   add(x1,x2)	weight: max{x2, x1}
a__fib1(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
    Usable rules: { 1..17 }
    Removed DPs: #1 #5..8 #10 #11 #15 #17..19 #22
Number of SCCs: 1, DPs: 13, edges: 68
	SCC { #2..4 #9 #12..14 #16 #20 #21 #23..25 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     s(x1)	weight: x1
#a__add(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}
a__add(x1,x2)	weight: max{x2, x1}
#a__fib1(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}
  fib1(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
   fib(x1)	weight: (/ 5 8) + x1
#mark(x1)	weight: (/ 1 8) + x1
     0()	weight: (/ 1 8)
   sel(x1,x2)	weight: max{(/ 1 4) + x2, x1}
#a__sel(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 8) + x1}
  mark(x1)	weight: x1
a__sel(x1,x2)	weight: max{(/ 1 4) + x2, x1}
#a__fib(x1)	weight: (/ 1 2) + x1
  cons(x1,x2)	weight: max{x2, (/ 1 8) + x1}
a__fib(x1)	weight: (/ 5 8) + x1
   add(x1,x2)	weight: max{x2, x1}
a__fib1(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
    Usable rules: { 1..17 }
    Removed DPs: #4
Number of SCCs: 1, DPs: 12, edges: 60
	SCC { #2 #3 #9 #12..14 #16 #20 #21 #23..25 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     s(x1)	weight: x1
#a__add(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}
a__add(x1,x2)	weight: max{x2, x1}
#a__fib1(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}
  fib1(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
   fib(x1)	weight: (/ 3 4) + x1
#mark(x1)	weight: (/ 1 8) + x1
     0()	weight: (/ 1 8)
   sel(x1,x2)	weight: max{(/ 3 8) + x2, (/ 1 4) + x1}
#a__sel(x1,x2)	weight: max{(/ 1 4) + x2, (/ 3 8) + x1}
  mark(x1)	weight: x1
a__sel(x1,x2)	weight: max{(/ 3 8) + x2, (/ 1 4) + x1}
#a__fib(x1)	weight: (/ 1 2) + x1
  cons(x1,x2)	weight: max{x2, (/ 1 4) + x1}
a__fib(x1)	weight: (/ 3 4) + x1
   add(x1,x2)	weight: max{x2, x1}
a__fib1(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
    Usable rules: { 1..17 }
    Removed DPs: #3 #21
Number of SCCs: 2, DPs: 9, edges: 33
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
     s(x1)	weight: x1	status: [x1]	precedence above:
#a__add(x1,x2)	weight: max{0, (/ 1 4) + x2}	status: [x2]	precedence above:
a__add(x1,x2)	weight: max{x2, x1}	status: [x1,x2]	precedence above: s add
#a__fib1(x1,x2)	weight: x2	status: []	precedence above:
  fib1(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}	status: []	precedence above: #a__sel cons a__fib1
   fib(x1)	weight: (/ 3 4)	status: []	precedence above: s fib1 sel #a__sel a__sel cons a__fib a__fib1
#mark(x1)	weight: (/ 1 4)	status: []	precedence above:
     0()	weight: (/ 1 4)	status: []	precedence above: s #a__sel mark cons
   sel(x1,x2)	weight: x2	status: []	precedence above: a__sel
#a__sel(x1,x2)	weight: (/ 1 2) + x2 + x1	status: [x1]	precedence above:
  mark(x1)	weight: x1	status: x1
a__sel(x1,x2)	weight: x2	status: []	precedence above: sel
#a__fib(x1)	weight: (/ 1 4)	status: []	precedence above:
  cons(x1,x2)	weight: max{x2, (/ 1 4) + x1}	status: []	precedence above: #a__sel
a__fib(x1)	weight: (/ 3 4)	status: []	precedence above: s fib1 fib sel #a__sel a__sel cons a__fib1
   add(x1,x2)	weight: max{x2, x1}	status: [x1,x2]	precedence above: s a__add
a__fib1(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}	status: []	precedence above: fib1 #a__sel cons
    Usable rules: { 1..17 }
    Removed DPs: #2
Number of SCCs: 1, DPs: 8, edges: 32
	SCC { #9 #12..14 #16 #23..25 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
     s(x1)	weight: x1	status: [x1]	precedence above: #a__add #mark
#a__add(x1,x2)	weight: max{x2, x1}	status: [x2,x1]	precedence above: #mark
a__add(x1,x2)	weight: max{x2, x1}	status: [x1,x2]	precedence above: s #a__add #mark add
#a__fib1(x1,x2)	weight: x2	status: []	precedence above:
  fib1(x1,x2)	weight: max{x2, x1}	status: []	precedence above: #a__sel cons a__fib1
   fib(x1)	weight: (/ 3 4)	status: []	precedence above: s #a__add fib1 #mark sel #a__sel a__sel cons a__fib a__fib1
#mark(x1)	weight: x1	status: [x1]	precedence above: #a__add
     0()	weight: (/ 1 4)	status: []	precedence above: s #a__add #mark #a__sel mark cons
   sel(x1,x2)	weight: (/ 1 4) + x2	status: []	precedence above: a__sel
#a__sel(x1,x2)	weight: (/ 1 2) + x2 + x1	status: [x1]	precedence above:
  mark(x1)	weight: x1	status: x1
a__sel(x1,x2)	weight: (/ 1 4) + x2	status: []	precedence above: sel
#a__fib(x1)	weight: (/ 1 4)	status: []	precedence above:
  cons(x1,x2)	weight: max{x2, x1}	status: []	precedence above: #a__sel
a__fib(x1)	weight: (/ 3 4)	status: []	precedence above: s #a__add fib1 fib #mark sel #a__sel a__sel cons a__fib1
   add(x1,x2)	weight: max{x2, x1}	status: [x1,x2]	precedence above: s #a__add a__add #mark
a__fib1(x1,x2)	weight: max{x2, x1}	status: []	precedence above: fib1 #a__sel cons
    Usable rules: { 1..17 }
    Removed DPs: #9 #12..14 #16 #23..25
Number of SCCs: 0, DPs: 0, edges: 0
YES