Input TRS:
    1: f(f(a(),x),a()) -> f(f(f(x,f(a(),a())),a()),a())
Number of strict rules: 1
Direct Order(PosReal,>,Poly) ... failed.
Freezing f
1: f❆2_a(x,a()) -> f(f(f(x,f❆1_a(a())),a()),a())
2: f(a(),_1) ->= f❆1_a(_1)
3: f(f❆1_a(_1),_2) ->= f❆2_a(_1,_2)
Number of strict rules: 1
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #f(f❆1_a(_1),_2) ->? #f❆2_a(_1,_2)
   #2: #f❆2_a(x,a()) -> #f(f(f(x,f❆1_a(a())),a()),a())
   #3: #f❆2_a(x,a()) -> #f(f(x,f❆1_a(a())),a())
   #4: #f❆2_a(x,a()) -> #f(x,f❆1_a(a()))
Number of SCCs: 1, DPs: 4, edges: 6
	SCC { #1..4 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     a()	weight: (/ 1 2)
f❆2_a(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
#f❆2_a(x1,x2)	weight: max{0, (/ 1 4) + x2}
     f(x1,x2)	weight: 0
   #f(x1,x2)	weight: max{0, (/ 1 4) + x2}
f❆1_a(x1)	weight: (/ 1 4)
    Usable rules: { }
    Removed DPs: #4
Number of SCCs: 1, DPs: 3, edges: 4
	SCC { #1..3 }
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❆2_a(x1,x2)	weight: (/ 1 4) + x2_2; 0
#f❆2_a(x1,x2)	weight: (/ 1 2) + x2_1 + x1_2; (/ 1 2) + x1_2
     f(x1,x2)	weight: (/ 1 4) + x2_2 + x1_2; 0
   #f(x1,x2)	weight: x2_1 + x1_1; x1_1
f❆1_a(x1)	weight: (/ 1 2) + x1_2; 0
    Usable rules: { 1..3 }
    Removed DPs: #3
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #1 #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 2)
f❆2_a(x1,x2)	weight: (/ 1 4); 0
#f❆2_a(x1,x2)	weight: (/ 1 2) + x2_1 + x2_2; (/ 1 2)
     f(x1,x2)	weight: (/ 1 4) + x1_2; 0
   #f(x1,x2)	weight: x2_1 + x2_2 + x1_1; (/ 1 4) + x1_1
f❆1_a(x1)	weight: (/ 3 4); 0
    Usable rules: { 1..3 }
    Removed DPs: #1 #2
Number of SCCs: 0, DPs: 0, edges: 0
YES