/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
Input TRS:
    1: even(0()) -> true()
    2: even(s(0())) -> false()
    3: even(s(s(x))) -> even(x)
    4: half(0()) -> 0()
    5: half(s(s(x))) -> s(half(x))
    6: plus(0(),y) -> y
    7: plus(s(x),y) -> s(plus(x,y))
    8: times(0(),y) -> 0()
    9: times(s(x),y) -> if_times(even(s(x)),s(x),y)
    10: if_times(true(),s(x),y) -> plus(times(half(s(x)),y),times(half(s(x)),y))
    11: if_times(false(),s(x),y) -> plus(y,times(x,y))
Number of strict rules: 11
Direct poly ... failed.
Freezing even half
1: even❆1_0() -> true()
2: even❆1_s(0()) -> false()
3: even❆1_s(s(x)) -> even(x)
4: half❆1_0() -> 0()
5: half❆1_s(s(x)) -> s(half(x))
6: plus(0(),y) -> y
7: plus(s(x),y) -> s(plus(x,y))
8: times(0(),y) -> 0()
9: times(s(x),y) -> if_times(even❆1_s(x),s(x),y)
10: if_times(true(),s(x),y) -> plus(times(half❆1_s(x),y),times(half❆1_s(x),y))
11: if_times(false(),s(x),y) -> plus(y,times(x,y))
12: half(0()) ->= half❆1_0()
13: half(s(_1)) ->= half❆1_s(_1)
14: even(0()) ->= even❆1_0()
15: even(s(_1)) ->= even❆1_s(_1)
Number of strict rules: 11
Direct poly ... failed.
Dependency Pairs:
   #1: #half(s(_1)) ->? #half❆1_s(_1)
   #2: #times(s(x),y) -> #if_times(even❆1_s(x),s(x),y)
   #3: #times(s(x),y) -> #even❆1_s(x)
   #4: #if_times(false(),s(x),y) -> #plus(y,times(x,y))
   #5: #if_times(false(),s(x),y) -> #times(x,y)
   #6: #half(0()) ->? #half❆1_0()
   #7: #even(0()) ->? #even❆1_0()
   #8: #plus(s(x),y) -> #plus(x,y)
   #9: #if_times(true(),s(x),y) -> #plus(times(half❆1_s(x),y),times(half❆1_s(x),y))
   #10: #if_times(true(),s(x),y) -> #times(half❆1_s(x),y)
   #11: #if_times(true(),s(x),y) -> #half❆1_s(x)
   #12: #if_times(true(),s(x),y) -> #times(half❆1_s(x),y)
   #13: #if_times(true(),s(x),y) -> #half❆1_s(x)
   #14: #half❆1_s(s(x)) -> #half(x)
   #15: #even❆1_s(s(x)) -> #even(x)
   #16: #even(s(_1)) ->? #even❆1_s(_1)
Number of SCCs: 4, DPs: 9
	SCC { #8 }
Sum... succeeded.
  s(x1)		w: (1 + x1)
  #if_times(x1,x2,x3)		w: (0)
  even❆1_0()		w: (0)
  #half❆1_s(x1)		w: (0)
  even(x1)		w: (0)
  #plus(x1,x2)		w: (x1)
  #even❆1_s(x1)		w: (0)
  false()		w: (0)
  #half(x1)		w: (0)
  even❆1_s(x1)		w: (0)
  true()		w: (0)
  #half❆1_0()		w: (0)
  half(x1)		w: (0)
  #even(x1)		w: (0)
  half❆1_s(x1)		w: (0)
  #times(x1,x2)		w: (0)
  0()		w: (0)
  times(x1,x2)		w: (0)
  #even❆1_0()		w: (0)
  if_times(x1,x2,x3)		w: (0)
  plus(x1,x2)		w: (0)
  half❆1_0()		w: (0)
    USABLE RULES: { }
    Removed DPs: #8
Number of SCCs: 3, DPs: 8
	SCC { #15 #16 }
Sum... succeeded.
  s(x1)		w: (3 + x1)
  #if_times(x1,x2,x3)		w: (0)
  even❆1_0()		w: (0)
  #half❆1_s(x1)		w: (0)
  even(x1)		w: (0)
  #plus(x1,x2)		w: (0)
  #even❆1_s(x1)		w: (x1)
  false()		w: (0)
  #half(x1)		w: (0)
  even❆1_s(x1)		w: (0)
  true()		w: (0)
  #half❆1_0()		w: (0)
  half(x1)		w: (0)
  #even(x1)		w: (2 + x1)
  half❆1_s(x1)		w: (0)
  #times(x1,x2)		w: (0)
  0()		w: (0)
  times(x1,x2)		w: (0)
  #even❆1_0()		w: (0)
  if_times(x1,x2,x3)		w: (0)
  plus(x1,x2)		w: (0)
  half❆1_0()		w: (0)
    USABLE RULES: { }
    Removed DPs: #15 #16
Number of SCCs: 2, DPs: 6
	SCC { #1 #14 }
Sum... succeeded.
  s(x1)		w: (1 + x1)
  #if_times(x1,x2,x3)		w: (0)
  even❆1_0()		w: (0)
  #half❆1_s(x1)		w: (x1)
  even(x1)		w: (0)
  #plus(x1,x2)		w: (0)
  #even❆1_s(x1)		w: (0)
  false()		w: (0)
  #half(x1)		w: (x1)
  even❆1_s(x1)		w: (0)
  true()		w: (0)
  #half❆1_0()		w: (0)
  half(x1)		w: (0)
  #even(x1)		w: (2)
  half❆1_s(x1)		w: (0)
  #times(x1,x2)		w: (0)
  0()		w: (0)
  times(x1,x2)		w: (0)
  #even❆1_0()		w: (0)
  if_times(x1,x2,x3)		w: (0)
  plus(x1,x2)		w: (0)
  half❆1_0()		w: (0)
    USABLE RULES: { }
    Removed DPs: #1 #14
Number of SCCs: 1, DPs: 4
	SCC { #2 #5 #10 #12 }
Sum... succeeded.
  s(x1)		w: (20540 + x1)
  #if_times(x1,x2,x3)		w: (x2)
  even❆1_0()		w: (20543)
  #half❆1_s(x1)		w: (0)
  even(x1)		w: (20542)
  #plus(x1,x2)		w: (0)
  #even❆1_s(x1)		w: (0)
  false()		w: (20542)
  #half(x1)		w: (0)
  even❆1_s(x1)		w: (20541)
  true()		w: (20544)
  #half❆1_0()		w: (0)
  half(x1)		w: (x1)
  #even(x1)		w: (2)
  half❆1_s(x1)		w: (1 + x1)
  #times(x1,x2)		w: (1 + x1)
  0()		w: (1)
  times(x1,x2)		w: (0)
  #even❆1_0()		w: (0)
  if_times(x1,x2,x3)		w: (0)
  plus(x1,x2)		w: (0)
  half❆1_0()		w: (1)
    USABLE RULES: { 4 5 12 13 }
    Removed DPs: #2 #5 #10 #12
Number of SCCs: 0, DPs: 0