Input TRS:
    1: -(x,|0|()) -> x
    2: -(s(x),s(y)) -> -(x,y)
    3: _lt_=(|0|(),y) -> true()
    4: _lt_=(s(x),|0|()) -> false()
    5: _lt_=(s(x),s(y)) -> _lt_=(x,y)
    6: if(true(),x,y) -> x
    7: if(false(),x,y) -> y
    8: perfectp(|0|()) -> false()
    9: perfectp(s(x)) -> f(x,s(|0|()),s(x),s(x))
    10: f(|0|(),y,|0|(),u) -> true()
    11: f(|0|(),y,s(z),u) -> false()
    12: f(s(x),|0|(),z,u) -> f(x,u,-(z,s(x)),u)
    13: f(s(x),s(y),z,u) -> if(_lt_=(x,y),f(s(x),-(y,x),z,u),f(x,u,z,u))
Number of strict rules: 13
Direct Order(PosReal,>,Poly) ... failed.
Freezing f
1: -(x,|0|()) -> x
2: -(s(x),s(y)) -> -(x,y)
3: _lt_=(|0|(),y) -> true()
4: _lt_=(s(x),|0|()) -> false()
5: _lt_=(s(x),s(y)) -> _lt_=(x,y)
6: if(true(),x,y) -> x
7: if(false(),x,y) -> y
8: perfectp(|0|()) -> false()
9: perfectp(s(x)) -> f(x,s(|0|()),s(x),s(x))
10: f❆1_|0|(y,|0|(),u) -> true()
11: f❆1_|0|(y,s(z),u) -> false()
12: f❆1_s(x,|0|(),z,u) -> f(x,u,-(z,s(x)),u)
13: f❆1_s(x,s(y),z,u) -> if(_lt_=(x,y),f❆1_s(x,-(y,x),z,u),f(x,u,z,u))
14: f(|0|(),_3,_4,_5) ->= f❆1_|0|(_3,_4,_5)
15: f(s(_1),_4,_5,_6) ->= f❆1_s(_1,_4,_5,_6)
Number of strict rules: 13
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #-(s(x),s(y)) -> #-(x,y)
   #2: #f❆1_s(x,s(y),z,u) -> #if(_lt_=(x,y),f❆1_s(x,-(y,x),z,u),f(x,u,z,u))
   #3: #f❆1_s(x,s(y),z,u) -> #_lt_=(x,y)
   #4: #f❆1_s(x,s(y),z,u) -> #f❆1_s(x,-(y,x),z,u)
   #5: #f❆1_s(x,s(y),z,u) -> #-(y,x)
   #6: #f❆1_s(x,s(y),z,u) -> #f(x,u,z,u)
   #7: #perfectp(s(x)) -> #f(x,s(|0|()),s(x),s(x))
   #8: #f❆1_s(x,|0|(),z,u) -> #f(x,u,-(z,s(x)),u)
   #9: #f❆1_s(x,|0|(),z,u) -> #-(z,s(x))
   #10: #f(|0|(),_3,_4,_5) ->? #f❆1_|0|(_3,_4,_5)
   #11: #_lt_=(s(x),s(y)) -> #_lt_=(x,y)
   #12: #f(s(_1),_4,_5,_6) ->? #f❆1_s(_1,_4,_5,_6)
Number of SCCs: 3, DPs: 6, edges: 10
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
#f❆1_s(x1,x2,x3,x4)	weight: 0
#f❆1_|0|(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 _lt_=(x1,x2)	weight: 0
#perfectp(x1)	weight: 0
 false()	weight: 0
  true()	weight: 0
f❆1_s(x1,x2,x3,x4)	weight: 0
     f(x1,x2,x3,x4)	weight: 0
f❆1_|0|(x1,x2,x3)	weight: 0
    if(x1,x2,x3)	weight: 0
     -(x1,x2)	weight: 0
   #f(x1,x2,x3,x4)	weight: 0
   #-(x1,x2)	weight: x2
  #if(x1,x2,x3)	weight: 0
#_lt_=(x1,x2)	weight: 0
perfectp(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #1
Number of SCCs: 2, DPs: 5, edges: 9
	SCC { #11 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
#f❆1_s(x1,x2,x3,x4)	weight: 0
#f❆1_|0|(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 _lt_=(x1,x2)	weight: 0
#perfectp(x1)	weight: 0
 false()	weight: 0
  true()	weight: 0
f❆1_s(x1,x2,x3,x4)	weight: 0
     f(x1,x2,x3,x4)	weight: 0
f❆1_|0|(x1,x2,x3)	weight: 0
    if(x1,x2,x3)	weight: 0
     -(x1,x2)	weight: 0
   #f(x1,x2,x3,x4)	weight: 0
   #-(x1,x2)	weight: 0
  #if(x1,x2,x3)	weight: 0
#_lt_=(x1,x2)	weight: x2
perfectp(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #11
Number of SCCs: 1, DPs: 4, edges: 8
	SCC { #4 #6 #8 #12 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
#f❆1_s(x1,x2,x3,x4)	weight: (/ 1 4) + x1
#f❆1_|0|(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 _lt_=(x1,x2)	weight: 0
#perfectp(x1)	weight: 0
 false()	weight: 0
  true()	weight: 0
f❆1_s(x1,x2,x3,x4)	weight: 0
     f(x1,x2,x3,x4)	weight: 0
f❆1_|0|(x1,x2,x3)	weight: 0
    if(x1,x2,x3)	weight: 0
     -(x1,x2)	weight: (/ 1 4) + x2
   #f(x1,x2,x3,x4)	weight: x1
   #-(x1,x2)	weight: 0
  #if(x1,x2,x3)	weight: 0
#_lt_=(x1,x2)	weight: 0
perfectp(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #6 #8 #12
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #4 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
#f❆1_s(x1,x2,x3,x4)	weight: (/ 1 4) + x2
#f❆1_|0|(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 _lt_=(x1,x2)	weight: 0
#perfectp(x1)	weight: 0
 false()	weight: 0
  true()	weight: 0
f❆1_s(x1,x2,x3,x4)	weight: 0
     f(x1,x2,x3,x4)	weight: 0
f❆1_|0|(x1,x2,x3)	weight: 0
    if(x1,x2,x3)	weight: 0
     -(x1,x2)	weight: (/ 1 4) + x1
   #f(x1,x2,x3,x4)	weight: 0
   #-(x1,x2)	weight: 0
  #if(x1,x2,x3)	weight: 0
#_lt_=(x1,x2)	weight: 0
perfectp(x1)	weight: 0
    Usable rules: { 1 2 }
    Removed DPs: #4
Number of SCCs: 0, DPs: 0, edges: 0
YES