Input TRS:
    1: g(x,0()) -> 0()
    2: g(d(),s(x)) -> s(s(g(d(),x)))
    3: g(h(),s(0())) -> 0()
    4: g(h(),s(s(x))) -> s(g(h(),x))
    5: double(x) -> g(d(),x)
    6: half(x) -> g(h(),x)
    7: f(s(x),y) -> f(half(s(x)),double(y))
    8: f(s(0()),y) -> y
    9: id(x) -> f(x,s(0()))
Number of strict rules: 9
Direct Order(PosReal,>,Poly) ... removes: 8 9
     h()	weight: 0
     d()	weight: 0
     s(x1)	weight: x1
     f(x1,x2)	weight: (/ 1 4) + x1 + x2
  half(x1)	weight: x1
     0()	weight: 0
double(x1)	weight: x1
    id(x1)	weight: (/ 1 2) + x1
     g(x1,x2)	weight: x1 + x2
Number of strict rules: 7
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #g(d(),s(x)) -> #g(d(),x)
   #2: #half(x) -> #g(h(),x)
   #3: #f(s(x),y) -> #f(half(s(x)),double(y))
   #4: #f(s(x),y) -> #half(s(x))
   #5: #f(s(x),y) -> #double(y)
   #6: #double(x) -> #g(d(),x)
   #7: #g(h(),s(s(x))) -> #g(h(),x)
Number of SCCs: 3, DPs: 3, edges: 3
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     d()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#half(x1)	weight: 0
     f(x1,x2)	weight: 0
  half(x1)	weight: 0
     0()	weight: 0
#double(x1)	weight: 0
double(x1)	weight: 0
   #f(x1,x2)	weight: 0
   #g(x1,x2)	weight: x2
    id(x1)	weight: 0
     g(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #1
Number of SCCs: 2, DPs: 2, edges: 2
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     d()	weight: 0
     s(x1)	weight: (/ 1 4) + x1
#half(x1)	weight: 0
     f(x1,x2)	weight: 0
  half(x1)	weight: 0
     0()	weight: 0
#double(x1)	weight: 0
double(x1)	weight: 0
   #f(x1,x2)	weight: 0
   #g(x1,x2)	weight: x2
    id(x1)	weight: 0
     g(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #7
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #3 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... succeeded.
     h()	weight: (/ 1 8)
     d()	weight: (/ 1 8)
     s(x1)	weight: max{0, (/ 3 8) + x1}
#half(x1)	weight: 0
     f(x1,x2)	weight: 0
  half(x1)	weight: max{0, (- (/ 1 8)) + x1}
     0()	weight: 0
#double(x1)	weight: 0
double(x1)	weight: (max (/ 1 8) 0)
   #f(x1,x2)	weight: max{0, (- (/ 1 8)) + x1}
   #g(x1,x2)	weight: 0
    id(x1)	weight: 0
     g(x1,x2)	weight: max{0, (- (/ 1 8)) + x2}
    Usable rules: { 1 3 4 6 }
    Removed DPs: #3
Number of SCCs: 0, DPs: 0, edges: 0
YES