Input TRS:
    1: f(s(x)) -> f(id_inc(c(x,x)))
    2: f(c(s(x),y)) -> g(c(x,y))
    3: g(c(s(x),y)) -> g(c(y,x))
    4: g(c(x,s(y))) -> g(c(y,x))
    5: g(c(x,x)) -> f(x)
    6: id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
    7: id_inc(s(x)) -> s(id_inc(x))
    8: id_inc(0()) -> 0()
    9: id_inc(0()) -> s(0())
Number of strict rules: 9
Direct Order(PosReal,>,Poly) ... failed.
Freezing g id_inc
1: f(s(x)) -> f(id_inc❆1_c(x,x))
2: f(c(s(x),y)) -> g❆1_c(x,y)
3: g❆1_c(s(x),y) -> g❆1_c(y,x)
4: g❆1_c(x,s(y)) -> g❆1_c(y,x)
5: g❆1_c(x,x) -> f(x)
6: id_inc❆1_c(x,y) -> c(id_inc(x),id_inc(y))
7: id_inc❆1_s(x) -> s(id_inc(x))
8: id_inc❆1_0() -> 0()
9: id_inc❆1_0() -> s(0())
10: id_inc(0()) ->= id_inc❆1_0()
11: id_inc(s(_1)) ->= id_inc❆1_s(_1)
12: id_inc(c(_1,_2)) ->= id_inc❆1_c(_1,_2)
13: g(c(_1,_2)) ->= g❆1_c(_1,_2)
Number of strict rules: 9
Direct Order(PosReal,>,Poly) ... removes: 13
     s(x1)	weight: 2 * x1
g❆1_c(x1,x2)	weight: x1 + x2
id_inc❆1_0()	weight: 0
     c(x1,x2)	weight: x1 + x2
     f(x1)	weight: x1
     0()	weight: 0
id_inc❆1_c(x1,x2)	weight: x1 + x2
id_inc❆1_s(x1)	weight: 2 * x1
id_inc(x1)	weight: x1
     g(x1)	weight: (/ 1 2) + x1
Number of strict rules: 9
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #f(c(s(x),y)) -> #g❆1_c(x,y)
   #2: #id_inc❆1_c(x,y) -> #id_inc(x)
   #3: #id_inc❆1_c(x,y) -> #id_inc(y)
   #4: #id_inc(s(_1)) ->? #id_inc❆1_s(_1)
   #5: #id_inc(c(_1,_2)) ->? #id_inc❆1_c(_1,_2)
   #6: #id_inc❆1_s(x) -> #id_inc(x)
   #7: #id_inc(0()) ->? #id_inc❆1_0()
   #8: #g❆1_c(x,x) -> #f(x)
   #9: #g❆1_c(s(x),y) -> #g❆1_c(y,x)
   #10: #f(s(x)) -> #f(id_inc❆1_c(x,x))
   #11: #f(s(x)) -> #id_inc❆1_c(x,x)
   #12: #g❆1_c(x,s(y)) -> #g❆1_c(y,x)
Number of SCCs: 2, DPs: 10, edges: 21
	SCC { #2..6 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
g❆1_c(x1,x2)	weight: 0
id_inc❆1_0()	weight: 0
#g❆1_c(x1,x2)	weight: 0
     c(x1,x2)	weight: (/ 1 2) + x1 + x2
     f(x1)	weight: 0
#id_inc(x1)	weight: x1
#id_inc❆1_0()	weight: 0
     0()	weight: 0
   #f(x1)	weight: 0
id_inc❆1_c(x1,x2)	weight: 0
id_inc❆1_s(x1)	weight: 0
#id_inc❆1_s(x1)	weight: (/ 1 4) + x1
id_inc(x1)	weight: 0
#id_inc❆1_c(x1,x2)	weight: (/ 1 4) + x1 + x2
     g(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #2..6
Number of SCCs: 1, DPs: 5, edges: 12
	SCC { #1 #8..10 #12 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... Order(PosReal,>,Sum-Sum; NegReal,≥,Sum)... Order(PosReal,>,MaxSum-Sum; NegReal,≥,Sum)... failed.
Removing edges: failed.
Finding a loop...  failed.
MAYBE