Input TRS:
    1: ack(0(),y) -> s(y)
    2: ack(s(x),0()) -> ack(x,s(0()))
    3: ack(s(x),s(y)) -> ack(x,ack(s(x),y))
    4: f(s(x),y) -> f(x,s(x))
    5: f(x,s(y)) -> f(y,x)
    6: f(x,y) -> ack(x,y)
    7: ack(s(x),y) -> f(x,x)
Number of strict rules: 7
Direct Order(PosReal,>,Poly) ... failed.
Freezing ack
1: ack❆1_0(y) -> s(y)
2: ack❆1_s(x,0()) -> ack(x,s(0()))
3: ack❆1_s(x,s(y)) -> ack(x,ack❆1_s(x,y))
4: f(s(x),y) -> f(x,s(x))
5: f(x,s(y)) -> f(y,x)
6: f(x,y) -> ack(x,y)
7: ack❆1_s(x,y) -> f(x,x)
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: #ack❆1_s(x,0()) -> #ack(x,s(0()))
   #2: #f(x,y) -> #ack(x,y)
   #3: #ack(s(_1),_2) ->? #ack❆1_s(_1,_2)
   #4: #ack❆1_s(x,y) -> #f(x,x)
   #5: #f(x,s(y)) -> #f(y,x)
   #6: #ack❆1_s(x,s(y)) -> #ack(x,ack❆1_s(x,y))
   #7: #ack❆1_s(x,s(y)) -> #ack❆1_s(x,y)
   #8: #ack(0(),_1) ->? #ack❆1_0(_1)
   #9: #f(s(x),y) -> #f(x,s(x))
Number of SCCs: 1, DPs: 8, edges: 20
	SCC { #1..7 #9 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
   ack(x1,x2)	weight: 0
#ack❆1_0(x1)	weight: 0
     f(x1,x2)	weight: max{0, (/ 1 8) + x2}
     0()	weight: (/ 1 8)
   #f(x1,x2)	weight: max{(/ 1 4) + x2, (/ 3 8) + x1}
ack❆1_0(x1)	weight: (/ 1 8)
 #ack(x1,x2)	weight: max{0, (/ 1 8) + x1}
ack❆1_s(x1,x2)	weight: 0
#ack❆1_s(x1,x2)	weight: max{0, (/ 1 2) + x1}
    Usable rules: { }
    Removed DPs: #1..6 #9
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 4) + x1
   ack(x1,x2)	weight: (/ 3 4) + x1
#ack❆1_0(x1)	weight: 0
     f(x1,x2)	weight: (/ 1 2)
     0()	weight: 0
   #f(x1,x2)	weight: 0
ack❆1_0(x1)	weight: 0
 #ack(x1,x2)	weight: 0
ack❆1_s(x1,x2)	weight: (/ 1 4) + x2
#ack❆1_s(x1,x2)	weight: x2
    Usable rules: { }
    Removed DPs: #7
Number of SCCs: 0, DPs: 0, edges: 0
YES