Input TRS:
    1: f(a()) -> g(h(a()))
    2: h(g(x)) -> g(h(f(x)))
    3: k(x,h(x),a()) -> h(x)
    4: k(f(x),y,x) -> f(x)
Number of strict rules: 4
Direct Order(PosReal,>,Poly) ... removes: 4 3
     a()	weight: 0
     h(x1)	weight: x1
     k(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + 2 * x3
     f(x1)	weight: x1
     g(x1)	weight: x1
Number of strict rules: 2
Direct Order(PosReal,>,Poly) ... failed.
Freezing h
1: f(a()) -> g(h❆1_a())
2: h❆1_g(x) -> g(h(f(x)))
5: h(a()) ->= h❆1_a()
6: h(g(_1)) ->= h❆1_g(_1)
Number of strict rules: 2
Direct Order(PosReal,>,Poly) ... removes: 5
     a()	weight: 0
     h(x1)	weight: (/ 1 4) + x1
h❆1_a()	weight: 0
     k(x1,x2,x3)	weight: x1 + x2 + x3
     f(x1)	weight: x1
h❆1_g(x1)	weight: (/ 1 4) + x1
     g(x1)	weight: x1
Number of strict rules: 2
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #h❆1_g(x) -> #h(f(x))
   #2: #h❆1_g(x) -> #f(x)
   #3: #h(g(_1)) ->? #h❆1_g(_1)
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #1 #3 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     a()	weight: (/ 3 4)
     h(x1)	weight: 0
h❆1_a()	weight: (/ 1 4)
     k(x1,x2,x3)	weight: 0
     f(x1)	weight: (/ 1 4) + x1
   #h(x1)	weight: x1
#h❆1_g(x1)	weight: (/ 1 2) + x1
   #f(x1)	weight: 0
h❆1_g(x1)	weight: 0
     g(x1)	weight: (/ 3 4) + x1
    Usable rules: { 1 }
    Removed DPs: #1 #3
Number of SCCs: 0, DPs: 0, edges: 0
YES