Input TRS:
    1: while(true(),s(s(s(i)))) -> while(gt(s(s(s(i))),s(|0|())),f(s(s(s(i)))))
    2: f(i) -> if(neq(i,s(s(|0|()))),i)
    3: gt(s(x),s(y)) -> gt(x,y)
    4: gt(s(x),|0|()) -> true()
    5: gt(|0|(),|0|()) -> false()
    6: gt(|0|(),s(y)) -> false()
    7: if(true(),i) -> plus(i,s(|0|()))
    8: if(false(),i) -> i
    9: neq(s(x),s(y)) -> neq(x,y)
    10: neq(|0|(),|0|()) -> false()
    11: neq(|0|(),s(y)) -> true()
    12: neq(s(x),|0|()) -> true()
    13: plus(s(x),y) -> plus(x,s(y))
    14: plus(|0|(),y) -> y
Number of strict rules: 14
Direct Order(PosReal,>,Poly) ... failed.
Freezing gt
1: while(true(),s(s(s(i)))) -> while(gt❆1_s(s(s(i)),s(|0|())),f(s(s(s(i)))))
2: f(i) -> if(neq(i,s(s(|0|()))),i)
3: gt❆1_s(x,s(y)) -> gt(x,y)
4: gt❆1_s(x,|0|()) -> true()
5: gt❆1_|0|(|0|()) -> false()
6: gt❆1_|0|(s(y)) -> false()
7: if(true(),i) -> plus(i,s(|0|()))
8: if(false(),i) -> i
9: neq(s(x),s(y)) -> neq(x,y)
10: neq(|0|(),|0|()) -> false()
11: neq(|0|(),s(y)) -> true()
12: neq(s(x),|0|()) -> true()
13: plus(s(x),y) -> plus(x,s(y))
14: plus(|0|(),y) -> y
15: gt(|0|(),_1) ->= gt❆1_|0|(_1)
16: gt(s(_1),_2) ->= gt❆1_s(_1,_2)
Number of strict rules: 14
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #f(i) -> #if(neq(i,s(s(|0|()))),i)
   #2: #f(i) -> #neq(i,s(s(|0|())))
   #3: #plus(s(x),y) -> #plus(x,s(y))
   #4: #neq(s(x),s(y)) -> #neq(x,y)
   #5: #if(true(),i) -> #plus(i,s(|0|()))
   #6: #gt(s(_1),_2) ->? #gt❆1_s(_1,_2)
   #7: #gt❆1_s(x,s(y)) -> #gt(x,y)
   #8: #while(true(),s(s(s(i)))) -> #while(gt❆1_s(s(s(i)),s(|0|())),f(s(s(s(i)))))
   #9: #while(true(),s(s(s(i)))) -> #gt❆1_s(s(s(i)),s(|0|()))
   #10: #while(true(),s(s(s(i)))) -> #f(s(s(s(i))))
   #11: #gt(|0|(),_1) ->? #gt❆1_|0|(_1)
Number of SCCs: 4, DPs: 5, edges: 5
	SCC { #4 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
   neq(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
    gt(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
 false()	weight: 0
#while(x1,x2)	weight: 0
  true()	weight: 0
     f(x1)	weight: 0
    if(x1,x2)	weight: 0
 while(x1,x2)	weight: 0
 #neq(x1,x2)	weight: x2
gt❆1_|0|(x1)	weight: 0
  #gt(x1,x2)	weight: 0
   #f(x1)	weight: 0
#gt❆1_s(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
  #if(x1,x2)	weight: 0
#gt❆1_|0|(x1)	weight: 0
gt❆1_s(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #4
Number of SCCs: 3, DPs: 4, edges: 4
	SCC { #3 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
   neq(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
    gt(x1,x2)	weight: 0
#plus(x1,x2)	weight: x1
 false()	weight: 0
#while(x1,x2)	weight: 0
  true()	weight: 0
     f(x1)	weight: 0
    if(x1,x2)	weight: 0
 while(x1,x2)	weight: 0
 #neq(x1,x2)	weight: 0
gt❆1_|0|(x1)	weight: 0
  #gt(x1,x2)	weight: 0
   #f(x1)	weight: 0
#gt❆1_s(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
  #if(x1,x2)	weight: 0
#gt❆1_|0|(x1)	weight: 0
gt❆1_s(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #3
Number of SCCs: 2, DPs: 3, edges: 3
	SCC { #6 #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
   neq(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
    gt(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
 false()	weight: 0
#while(x1,x2)	weight: 0
  true()	weight: 0
     f(x1)	weight: 0
    if(x1,x2)	weight: 0
 while(x1,x2)	weight: 0
 #neq(x1,x2)	weight: 0
gt❆1_|0|(x1)	weight: 0
  #gt(x1,x2)	weight: x1
   #f(x1)	weight: 0
#gt❆1_s(x1,x2)	weight: (/ 1 4) + x1
  plus(x1,x2)	weight: 0
  #if(x1,x2)	weight: 0
#gt❆1_|0|(x1)	weight: 0
gt❆1_s(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #6 #7
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #8 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... Order(PosReal,>,Sum-Sum; NegReal,≥,Sum)... Order(PosReal,>,MaxSum-Sum; NegReal,≥,Sum)... failed.
Removing edges: failed.
Finding a loop...  failed.
MAYBE