Input TRS:
    1: lt(|0|(),s(x)) -> true()
    2: lt(x,|0|()) -> false()
    3: lt(s(x),s(y)) -> lt(x,y)
    4: fibo(|0|()) -> fib(|0|())
    5: fibo(s(|0|())) -> fib(s(|0|()))
    6: fibo(s(s(x))) -> sum(fibo(s(x)),fibo(x))
    7: fib(|0|()) -> s(|0|())
    8: fib(s(|0|())) -> s(|0|())
    9: fib(s(s(x))) -> if(true(),|0|(),s(s(x)),|0|(),|0|())
    10: if(true(),c,s(s(x)),a,b) -> if(lt(s(c),s(s(x))),s(c),s(s(x)),b,c)
    11: if(false(),c,s(s(x)),a,b) -> sum(fibo(a),fibo(b))
    12: sum(x,|0|()) -> x
    13: sum(x,s(y)) -> s(sum(x,y))
Number of strict rules: 13
Direct Order(PosReal,>,Poly) ... failed.
Freezing fibo if fib
1: lt(|0|(),s(x)) -> true()
2: lt(x,|0|()) -> false()
3: lt(s(x),s(y)) -> lt(x,y)
4: fibo❆1_|0|() -> fib❆1_|0|()
5: fibo❆1_s(|0|()) -> fib❆1_s(|0|())
6: fibo❆1_s(s(x)) -> sum(fibo❆1_s(x),fibo(x))
7: fib❆1_|0|() -> s(|0|())
8: fib❆1_s(|0|()) -> s(|0|())
9: fib❆1_s(s(x)) -> if❆1_true(|0|(),s(s(x)),|0|(),|0|())
10: if❆1_true(c,s(s(x)),a,b) -> if(lt(s(c),s(s(x))),s(c),s(s(x)),b,c)
11: if❆1_false(c,s(s(x)),a,b) -> sum(fibo(a),fibo(b))
12: sum(x,|0|()) -> x
13: sum(x,s(y)) -> s(sum(x,y))
14: fib(|0|()) ->= fib❆1_|0|()
15: fib(s(_1)) ->= fib❆1_s(_1)
16: if(false(),_4,_5,_6,_7) ->= if❆1_false(_4,_5,_6,_7)
17: if(true(),_4,_5,_6,_7) ->= if❆1_true(_4,_5,_6,_7)
18: fibo(|0|()) ->= fibo❆1_|0|()
19: fibo(s(_1)) ->= fibo❆1_s(_1)
Number of strict rules: 13
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #fibo❆1_s(s(x)) -> #sum(fibo❆1_s(x),fibo(x))
   #2: #fibo❆1_s(s(x)) -> #fibo❆1_s(x)
   #3: #fibo❆1_s(s(x)) -> #fibo(x)
   #4: #sum(x,s(y)) -> #sum(x,y)
   #5: #fib❆1_s(s(x)) -> #if❆1_true(|0|(),s(s(x)),|0|(),|0|())
   #6: #if❆1_false(c,s(s(x)),a,b) -> #sum(fibo(a),fibo(b))
   #7: #if❆1_false(c,s(s(x)),a,b) -> #fibo(a)
   #8: #if❆1_false(c,s(s(x)),a,b) -> #fibo(b)
   #9: #fib(|0|()) ->? #fib❆1_|0|()
   #10: #if❆1_true(c,s(s(x)),a,b) -> #if(lt(s(c),s(s(x))),s(c),s(s(x)),b,c)
   #11: #if❆1_true(c,s(s(x)),a,b) -> #lt(s(c),s(s(x)))
   #12: #fibo❆1_s(|0|()) -> #fib❆1_s(|0|())
   #13: #if(true(),_4,_5,_6,_7) ->? #if❆1_true(_4,_5,_6,_7)
   #14: #fibo(s(_1)) ->? #fibo❆1_s(_1)
   #15: #if(false(),_4,_5,_6,_7) ->? #if❆1_false(_4,_5,_6,_7)
   #16: #lt(s(x),s(y)) -> #lt(x,y)
   #17: #fib(s(_1)) ->? #fib❆1_s(_1)
   #18: #fibo❆1_|0|() -> #fib❆1_|0|()
   #19: #fibo(|0|()) ->? #fibo❆1_|0|()
Number of SCCs: 4, DPs: 7, edges: 9
	SCC { #4 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
fib❆1_|0|()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#fib❆1_s(x1)	weight: 0
  #lt(x1,x2)	weight: 0
#fibo❆1_|0|()	weight: 0
  fibo(x1)	weight: 0
 false()	weight: 0
fibo❆1_|0|()	weight: 0
if❆1_false(x1,x2,x3,x4)	weight: 0
#if❆1_false(x1,x2,x3,x4)	weight: 0
#fibo❆1_s(x1)	weight: 0
#fibo(x1)	weight: 0
  true()	weight: 0
   sum(x1,x2)	weight: 0
#fib❆1_|0|()	weight: 0
   fib(x1)	weight: 0
    if(x1,x2,x3,x4,x5)	weight: 0
if❆1_true(x1,x2,x3,x4)	weight: 0
fibo❆1_s(x1)	weight: 0
fib❆1_s(x1)	weight: 0
  #if(x1,x2,x3,x4,x5)	weight: 0
#if❆1_true(x1,x2,x3,x4)	weight: 0
 #fib(x1)	weight: 0
 #sum(x1,x2)	weight: x2
    lt(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #4
Number of SCCs: 3, DPs: 6, edges: 8
	SCC { #16 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
fib❆1_|0|()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#fib❆1_s(x1)	weight: 0
  #lt(x1,x2)	weight: x2
#fibo❆1_|0|()	weight: 0
  fibo(x1)	weight: 0
 false()	weight: 0
fibo❆1_|0|()	weight: 0
if❆1_false(x1,x2,x3,x4)	weight: 0
#if❆1_false(x1,x2,x3,x4)	weight: 0
#fibo❆1_s(x1)	weight: 0
#fibo(x1)	weight: 0
  true()	weight: 0
   sum(x1,x2)	weight: 0
#fib❆1_|0|()	weight: 0
   fib(x1)	weight: 0
    if(x1,x2,x3,x4,x5)	weight: 0
if❆1_true(x1,x2,x3,x4)	weight: 0
fibo❆1_s(x1)	weight: 0
fib❆1_s(x1)	weight: 0
  #if(x1,x2,x3,x4,x5)	weight: 0
#if❆1_true(x1,x2,x3,x4)	weight: 0
 #fib(x1)	weight: 0
 #sum(x1,x2)	weight: 0
    lt(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #16
Number of SCCs: 2, DPs: 5, edges: 7
	SCC { #2 #3 #14 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
fib❆1_|0|()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#fib❆1_s(x1)	weight: 0
  #lt(x1,x2)	weight: 0
#fibo❆1_|0|()	weight: 0
  fibo(x1)	weight: 0
 false()	weight: 0
fibo❆1_|0|()	weight: 0
if❆1_false(x1,x2,x3,x4)	weight: 0
#if❆1_false(x1,x2,x3,x4)	weight: 0
#fibo❆1_s(x1)	weight: x1
#fibo(x1)	weight: x1
  true()	weight: 0
   sum(x1,x2)	weight: 0
#fib❆1_|0|()	weight: 0
   fib(x1)	weight: 0
    if(x1,x2,x3,x4,x5)	weight: 0
if❆1_true(x1,x2,x3,x4)	weight: 0
fibo❆1_s(x1)	weight: 0
fib❆1_s(x1)	weight: 0
  #if(x1,x2,x3,x4,x5)	weight: 0
#if❆1_true(x1,x2,x3,x4)	weight: 0
 #fib(x1)	weight: 0
 #sum(x1,x2)	weight: 0
    lt(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2 #3 #14
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #10 #13 }
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