Input TRS:
    1: plus(s(s(x)),y) -> s(plus(x,s(y)))
    2: plus(x,s(s(y))) -> s(plus(s(x),y))
    3: plus(s(0()),y) -> s(y)
    4: plus(0(),y) -> y
    5: ack(0(),y) -> s(y)
    6: ack(s(x),0()) -> ack(x,s(0()))
    7: ack(s(x),s(y)) -> ack(x,plus(y,ack(s(x),y)))
Number of strict rules: 7
Direct Order(PosReal,>,Poly) ... failed.
Freezing ack
1: plus(s(s(x)),y) -> s(plus(x,s(y)))
2: plus(x,s(s(y))) -> s(plus(s(x),y))
3: plus(s(0()),y) -> s(y)
4: plus(0(),y) -> y
5: ack❆1_0(y) -> s(y)
6: ack❆1_s(x,0()) -> ack(x,s(0()))
7: ack❆1_s(x,s(y)) -> ack(x,plus(y,ack❆1_s(x,y)))
8: ack(0(),_1) ->= ack❆1_0(_1)
9: ack(s(_1),_2) ->= ack❆1_s(_1,_2)
Number of strict rules: 7
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #plus(x,s(s(y))) -> #plus(s(x),y)
   #2: #ack❆1_s(x,0()) -> #ack(x,s(0()))
   #3: #ack(s(_1),_2) ->? #ack❆1_s(_1,_2)
   #4: #ack❆1_s(x,s(y)) -> #ack(x,plus(y,ack❆1_s(x,y)))
   #5: #ack❆1_s(x,s(y)) -> #plus(y,ack❆1_s(x,y))
   #6: #ack❆1_s(x,s(y)) -> #ack❆1_s(x,y)
   #7: #plus(s(s(x)),y) -> #plus(x,s(y))
   #8: #ack(0(),_1) ->? #ack❆1_0(_1)
Number of SCCs: 2, DPs: 6, edges: 12
	SCC { #1 #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 4) + x1
   ack(x1,x2)	weight: 0
#plus(x1,x2)	weight: x1 + x2
#ack❆1_0(x1)	weight: 0
     0()	weight: 0
ack❆1_0(x1)	weight: 0
  plus(x1,x2)	weight: 0
 #ack(x1,x2)	weight: 0
ack❆1_s(x1,x2)	weight: 0
#ack❆1_s(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #1 #7
Number of SCCs: 1, DPs: 4, edges: 8
	SCC { #2..4 #6 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 4) + x1
   ack(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
#ack❆1_0(x1)	weight: 0
     0()	weight: 0
ack❆1_0(x1)	weight: (/ 1 8) + x1
  plus(x1,x2)	weight: (/ 1 8) + x2
 #ack(x1,x2)	weight: x1
ack❆1_s(x1,x2)	weight: (/ 3 8)
#ack❆1_s(x1,x2)	weight: (/ 1 8) + x1
    Usable rules: { }
    Removed DPs: #2..4
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #6 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 4) + x1
   ack(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
#ack❆1_0(x1)	weight: 0
     0()	weight: 0
ack❆1_0(x1)	weight: (/ 1 8) + x1
  plus(x1,x2)	weight: (/ 1 8) + x2
 #ack(x1,x2)	weight: 0
ack❆1_s(x1,x2)	weight: (/ 3 8)
#ack❆1_s(x1,x2)	weight: (/ 1 8) + x2
    Usable rules: { }
    Removed DPs: #6
Number of SCCs: 0, DPs: 0, edges: 0
YES