Input TRS:
    1: a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
    2: a(a(x)) -> f(b(),a(f(a(x),b())))
    3: f(a(x),b()) -> f(b(),a(x))
Number of strict rules: 3
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a(a(x)) -> #f(b(),a(f(a(x),b())))
   #2: #a(a(x)) -> #a(f(a(x),b()))
   #3: #a(a(x)) -> #f(a(x),b())
   #4: #f(a(x),b()) -> #f(b(),a(x))
   #5: #a(a(f(b(),a(x)))) -> #f(a(a(a(x))),b())
   #6: #a(a(f(b(),a(x)))) -> #a(a(a(x)))
   #7: #a(a(f(b(),a(x)))) -> #a(a(x))
Number of SCCs: 1, DPs: 2, edges: 4
	SCC { #6 #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1)	weight: (/ 1 4) + x1
     b()	weight: 0
     f(x1,x2)	weight: x1 + x2
   #f(x1,x2)	weight: 0
   #a(x1)	weight: x1
    Usable rules: { 1..3 }
    Removed DPs: #7
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #6 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... succeeded.
     a(x1)	weight: (/ 1 4) + x1_2; (/ 1 8) + x1_1
     b()	weight: (/ 1 8); 0
     f(x1,x2)	weight: (/ 1 8); (/ 1 8) + x2_2 + x1_2
   #f(x1,x2)	weight: 0; 0
   #a(x1)	weight: x1_1; 0
    Usable rules: { 1..3 }
    Removed DPs: #6
Number of SCCs: 0, DPs: 0, edges: 0
YES