Input TRS:
    1: app(app(map(),f),nil()) -> nil()
    2: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
    3: app(app(app(comp(),f),g),x) -> app(f,app(g,x))
    4: app(twice(),f) -> app(app(comp(),f),f)
Number of strict rules: 4
Direct Order(PosReal,>,Poly) ... failed.
Freezing app
1: app❆2_map(f,nil()) -> nil()
2: app❆2_map(f,app❆2_cons(x,xs)) -> app❆2_cons(app(f,x),app❆2_map(f,xs))
3: app❆3_comp(f,g,x) -> app(f,app(g,x))
4: app❆1_twice(f) -> app❆2_comp(f,f)
5: app(cons(),_1) ->= app❆1_cons(_1)
6: app(app❆1_cons(_1),_2) ->= app❆2_cons(_1,_2)
7: app(twice(),_1) ->= app❆1_twice(_1)
8: app(map(),_1) ->= app❆1_map(_1)
9: app(app❆1_map(_1),_2) ->= app❆2_map(_1,_2)
10: app(comp(),_1) ->= app❆1_comp(_1)
11: app(app❆1_comp(_1),_2) ->= app❆2_comp(_1,_2)
12: app(app❆2_comp(_1,_2),_3) ->= app❆3_comp(_1,_2,_3)
Number of strict rules: 4
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #app❆2_map(f,app❆2_cons(x,xs)) -> #app(f,x)
   #2: #app❆2_map(f,app❆2_cons(x,xs)) -> #app❆2_map(f,xs)
   #3: #app(app❆1_map(_1),_2) ->? #app❆2_map(_1,_2)
   #4: #app(app❆2_comp(_1,_2),_3) ->? #app❆3_comp(_1,_2,_3)
   #5: #app(twice(),_1) ->? #app❆1_twice(_1)
   #6: #app❆3_comp(f,g,x) -> #app(f,app(g,x))
   #7: #app❆3_comp(f,g,x) -> #app(g,x)
Number of SCCs: 1, DPs: 6, edges: 12
	SCC { #1..4 #6 #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#app❆2_map(x1,x2)	weight: (/ 1 4) + x1
app❆1_map(x1)	weight: (/ 1 2) + x1
 twice()	weight: 0
app❆2_comp(x1,x2)	weight: (/ 3 4) + x1 + x2
app❆2_map(x1,x2)	weight: (/ 1 2) + x1
app❆1_comp(x1)	weight: (/ 1 2) + x1
   nil()	weight: 0
 #app(x1,x2)	weight: x1
   map()	weight: 0
#app❆3_comp(x1,x2,x3)	weight: (/ 1 4) + x1 + x2
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 1 2) + x1 + x2
#app❆1_twice(x1)	weight: 0
app❆3_comp(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + x3
app❆1_cons(x1)	weight: (/ 1 2) + x1
app❆1_twice(x1)	weight: (/ 1 2)
  comp()	weight: 0
   app(x1,x2)	weight: (/ 1 4)
    Usable rules: { }
    Removed DPs: #1 #3 #4 #6 #7
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#app❆2_map(x1,x2)	weight: (/ 1 4) + x2
app❆1_map(x1)	weight: (/ 1 2) + x1
 twice()	weight: 0
app❆2_comp(x1,x2)	weight: (/ 3 4) + x1 + x2
app❆2_map(x1,x2)	weight: (/ 1 2) + x1
app❆1_comp(x1)	weight: (/ 1 2) + x1
   nil()	weight: 0
 #app(x1,x2)	weight: 0
   map()	weight: 0
#app❆3_comp(x1,x2,x3)	weight: (/ 1 4)
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 1 2) + x1 + x2
#app❆1_twice(x1)	weight: 0
app❆3_comp(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + x3
app❆1_cons(x1)	weight: (/ 1 2) + x1
app❆1_twice(x1)	weight: (/ 1 2)
  comp()	weight: 0
   app(x1,x2)	weight: (/ 1 4)
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 0, DPs: 0, edges: 0
YES