Input TRS:
    1: t(N) -> cs(r(q(N)),nt(ns(N)))
    2: q(0()) -> 0()
    3: q(s(X)) -> s(p(q(X),d(X)))
    4: d(0()) -> 0()
    5: d(s(X)) -> s(s(d(X)))
    6: p(0(),X) -> X
    7: p(X,0()) -> X
    8: p(s(X),s(Y)) -> s(s(p(X,Y)))
    9: f(0(),X) -> nil()
    10: f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
    11: t(X) -> nt(X)
    12: s(X) -> ns(X)
    13: f(X1,X2) -> nf(X1,X2)
    14: a(nt(X)) -> t(a(X))
    15: a(ns(X)) -> s(a(X))
    16: a(nf(X1,X2)) -> f(a(X1),a(X2))
    17: a(X) -> X
Number of strict rules: 17
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a(nt(X)) -> #t(a(X))
   #2: #a(nt(X)) -> #a(X)
   #3: #f(s(X),cs(Y,Z)) -> #a(Z)
   #4: #d(s(X)) -> #s(s(d(X)))
   #5: #d(s(X)) -> #s(d(X))
   #6: #d(s(X)) -> #d(X)
   #7: #a(nf(X1,X2)) -> #f(a(X1),a(X2))
   #8: #a(nf(X1,X2)) -> #a(X1)
   #9: #a(nf(X1,X2)) -> #a(X2)
   #10: #q(s(X)) -> #s(p(q(X),d(X)))
   #11: #q(s(X)) -> #p(q(X),d(X))
   #12: #q(s(X)) -> #q(X)
   #13: #q(s(X)) -> #d(X)
   #14: #t(N) -> #q(N)
   #15: #p(s(X),s(Y)) -> #s(s(p(X,Y)))
   #16: #p(s(X),s(Y)) -> #s(p(X,Y))
   #17: #p(s(X),s(Y)) -> #p(X,Y)
   #18: #a(ns(X)) -> #s(a(X))
   #19: #a(ns(X)) -> #a(X)
Number of SCCs: 4, DPs: 9, edges: 29
	SCC { #6 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1)	weight: 0
    nf(x1,x2)	weight: 0
     q(x1)	weight: 0
     d(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
     r(x1)	weight: 0
     t(x1)	weight: 0
    ns(x1)	weight: 0
   #p(x1,x2)	weight: 0
    cs(x1,x2)	weight: 0
     f(x1,x2)	weight: 0
     p(x1,x2)	weight: 0
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
   #d(x1)	weight: x1
   #f(x1,x2)	weight: 0
    nt(x1)	weight: 0
   #a(x1)	weight: 0
   #q(x1)	weight: 0
   #t(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #6
Number of SCCs: 3, DPs: 8, edges: 28
	SCC { #12 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1)	weight: 0
    nf(x1,x2)	weight: 0
     q(x1)	weight: 0
     d(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
     r(x1)	weight: 0
     t(x1)	weight: 0
    ns(x1)	weight: 0
   #p(x1,x2)	weight: 0
    cs(x1,x2)	weight: 0
     f(x1,x2)	weight: 0
     p(x1,x2)	weight: 0
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
   #d(x1)	weight: 0
   #f(x1,x2)	weight: 0
    nt(x1)	weight: 0
   #a(x1)	weight: 0
   #q(x1)	weight: x1
   #t(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #12
Number of SCCs: 2, DPs: 7, edges: 27
	SCC { #17 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1)	weight: 0
    nf(x1,x2)	weight: 0
     q(x1)	weight: 0
     d(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
     r(x1)	weight: 0
     t(x1)	weight: 0
    ns(x1)	weight: 0
   #p(x1,x2)	weight: x2
    cs(x1,x2)	weight: 0
     f(x1,x2)	weight: 0
     p(x1,x2)	weight: 0
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
   #d(x1)	weight: 0
   #f(x1,x2)	weight: 0
    nt(x1)	weight: 0
   #a(x1)	weight: 0
   #q(x1)	weight: 0
   #t(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #17
Number of SCCs: 1, DPs: 6, edges: 26
	SCC { #2 #3 #7..9 #19 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1)	weight: x1
    nf(x1,x2)	weight: (/ 1 2) + x1 + x2
     q(x1)	weight: (/ 1 4)
     d(x1)	weight: 0
     s(x1)	weight: x1
     r(x1)	weight: (/ 1 4)
     t(x1)	weight: (/ 1 4) + x1
    ns(x1)	weight: x1
   #p(x1,x2)	weight: 0
    cs(x1,x2)	weight: x2
     f(x1,x2)	weight: (/ 1 2) + x1 + x2
     p(x1,x2)	weight: x1 + x2
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
   #d(x1)	weight: 0
   #f(x1,x2)	weight: (/ 1 4) + x2
    nt(x1)	weight: (/ 1 4) + x1
   #a(x1)	weight: x1
   #q(x1)	weight: 0
   #t(x1)	weight: 0
    Usable rules: { 1..17 }
    Removed DPs: #2 #3 #7..9
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #19 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1)	weight: x1
    nf(x1,x2)	weight: (/ 1 4) + x1 + x2
     q(x1)	weight: (/ 1 2)
     d(x1)	weight: x1
     s(x1)	weight: (/ 1 4) + x1
     r(x1)	weight: (/ 1 4)
     t(x1)	weight: (/ 1 4) + x1
    ns(x1)	weight: (/ 1 4) + x1
   #p(x1,x2)	weight: 0
    cs(x1,x2)	weight: 0
     f(x1,x2)	weight: (/ 1 4) + x1 + x2
     p(x1,x2)	weight: (/ 1 4)
     0()	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
   #d(x1)	weight: 0
   #f(x1,x2)	weight: (/ 1 4) + x2
    nt(x1)	weight: (/ 1 4) + x1
   #a(x1)	weight: x1
   #q(x1)	weight: 0
   #t(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #19
Number of SCCs: 0, DPs: 0, edges: 0
YES