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