Input TRS:
    1: f(c(s(x),y)) -> f(c(x,s(y)))
    2: f(c(s(x),s(y))) -> g(c(x,y))
    3: g(c(x,s(y))) -> g(c(s(x),y))
    4: g(c(s(x),s(y))) -> f(c(x,y))
    5: rand(x) ->= x
    6: rand(x) ->= rand(s(x))
Number of strict rules: 4
Direct Order(PosReal,>,Poly) ... removes: 5
     s(x1)	weight: x1
     c(x1,x2)	weight: (/ 1 4) + 2 * x2 + x1
     f(x1)	weight: x1
  rand(x1)	weight: (/ 1 4) + x1
     g(x1)	weight: x1
Number of strict rules: 4
Direct Order(PosReal,>,Poly) ... failed.
Freezing f g
1: f❆1_c(s(x),y) -> f❆1_c(x,s(y))
2: f❆1_c(s(x),s(y)) -> g❆1_c(x,y)
3: g❆1_c(x,s(y)) -> g❆1_c(s(x),y)
4: g❆1_c(s(x),s(y)) -> f❆1_c(x,y)
6: rand(x) ->= rand(s(x))
7: g(c(_1,_2)) ->= g❆1_c(_1,_2)
8: f(c(_1,_2)) ->= f❆1_c(_1,_2)
Number of strict rules: 4
Direct Order(PosReal,>,Poly) ... removes: 8 7
     s(x1)	weight: x1
g❆1_c(x1,x2)	weight: 2 * x2 + 2 * x1
     c(x1,x2)	weight: (/ 1 4) + x2 + x1
f❆1_c(x1,x2)	weight: 2 * x2 + 2 * x1
     f(x1)	weight: 2 * x1
  rand(x1)	weight: 2 * x1
     g(x1)	weight: 2 * x1
Number of strict rules: 4
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #f❆1_c(s(x),s(y)) -> #g❆1_c(x,y)
   #2: #g❆1_c(x,s(y)) -> #g❆1_c(s(x),y)
   #3: #f❆1_c(s(x),y) -> #f❆1_c(x,s(y))
   #4: #g❆1_c(s(x),s(y)) -> #f❆1_c(x,y)
Number of SCCs: 1, DPs: 4, edges: 8
	SCC { #1..4 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
g❆1_c(x1,x2)	weight: 0
#g❆1_c(x1,x2)	weight: (/ 1 2) + x1 + x2
     c(x1,x2)	weight: 0
f❆1_c(x1,x2)	weight: 0
     f(x1)	weight: 0
  rand(x1)	weight: 0
#f❆1_c(x1,x2)	weight: x1 + x2
     g(x1)	weight: 0
    Removed DPs: #1 #4
Number of SCCs: 2, DPs: 2, edges: 2
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
g❆1_c(x1,x2)	weight: 0
#g❆1_c(x1,x2)	weight: (/ 1 2) + x2
     c(x1,x2)	weight: 0
f❆1_c(x1,x2)	weight: 0
     f(x1)	weight: 0
  rand(x1)	weight: 0
#f❆1_c(x1,x2)	weight: 0
     g(x1)	weight: 0
    Removed DPs: #2
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #3 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
g❆1_c(x1,x2)	weight: 0
#g❆1_c(x1,x2)	weight: (/ 1 2)
     c(x1,x2)	weight: 0
f❆1_c(x1,x2)	weight: 0
     f(x1)	weight: 0
  rand(x1)	weight: 0
#f❆1_c(x1,x2)	weight: x1
     g(x1)	weight: 0
    Removed DPs: #3
Number of SCCs: 0, DPs: 0, edges: 0
YES