Input TRS:
    1: f(a(),f(a(),x)) -> f(a(),f(x,f(f(a(),a()),a())))
Number of strict rules: 1
Direct Order(PosReal,>,Poly) ... failed.
Freezing f
1: f❆1_a(f❆1_a(x)) -> f❆1_a(f(x,f(f❆1_a(a()),a())))
2: f(a(),_1) ->= f❆1_a(_1)
Number of strict rules: 1
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #f(a(),_1) ->? #f❆1_a(_1)
   #2: #f❆1_a(f❆1_a(x)) -> #f❆1_a(f(x,f(f❆1_a(a()),a())))
   #3: #f❆1_a(f❆1_a(x)) -> #f(x,f(f❆1_a(a()),a()))
   #4: #f❆1_a(f❆1_a(x)) -> #f(f❆1_a(a()),a())
   #5: #f❆1_a(f❆1_a(x)) -> #f❆1_a(a())
Number of SCCs: 1, DPs: 3, edges: 5
	SCC { #1..3 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... succeeded.
     a()	weight: (/ 1 2)
     f(x1,x2)	weight: max{0, (- (/ 3 4)) + x2 + x1}
#f❆1_a(x1)	weight: max{0, (- (/ 1 8)) + x1}
   #f(x1,x2)	weight: max{0, (- (/ 3 8)) + x2 + x1}
f❆1_a(x1)	weight: max{0, (- (/ 1 4)) + x1}
    Usable rules: { 1 2 }
    Removed DPs: #1
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... succeeded.
     a()	weight: (/ 1 4); (/ 1 4)
     f(x1,x2)	weight: 0; x2_1 + x1_1
#f❆1_a(x1)	weight: x1_2; 0
   #f(x1,x2)	weight: 0; 0
f❆1_a(x1)	weight: 0; (/ 1 4) + x1_1
    Usable rules: { 1 2 }
    Removed DPs: #2
Number of SCCs: 0, DPs: 0, edges: 0
YES