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