Problem ICFP 2010 58301

Tool CaT

Execution TimeUnknown
Answer
YES(?,O(n^1))
InputICFP 2010 58301

stdout:

YES(?,O(n^1))

Problem:
 0(0(1(x1))) -> 1(2(0(2(0(x1)))))
 0(0(1(x1))) -> 2(0(2(1(0(x1)))))
 0(0(1(x1))) -> 2(1(0(2(0(x1)))))
 0(0(1(x1))) -> 1(1(2(0(2(0(x1))))))
 0(0(1(x1))) -> 1(2(0(2(0(2(x1))))))
 0(0(1(x1))) -> 1(2(0(2(0(3(x1))))))
 0(0(1(x1))) -> 1(2(0(2(0(4(x1))))))
 0(0(1(x1))) -> 1(2(0(2(2(0(x1))))))
 0(0(1(x1))) -> 1(2(0(2(3(0(x1))))))
 0(0(1(x1))) -> 1(2(0(2(5(0(x1))))))
 0(0(1(x1))) -> 1(2(0(3(2(0(x1))))))
 0(0(1(x1))) -> 1(2(1(0(2(0(x1))))))
 0(0(1(x1))) -> 1(2(2(0(2(0(x1))))))
 0(0(1(x1))) -> 1(2(2(0(3(0(x1))))))
 0(0(1(x1))) -> 1(2(2(0(4(0(x1))))))
 0(0(1(x1))) -> 1(2(3(0(2(0(x1))))))
 0(0(1(x1))) -> 1(2(5(0(2(0(x1))))))
 0(0(1(x1))) -> 1(3(2(0(2(0(x1))))))
 0(0(1(x1))) -> 2(0(1(2(0(2(x1))))))
 0(0(1(x1))) -> 2(0(1(2(2(0(x1))))))
 0(0(1(x1))) -> 2(0(1(3(0(2(x1))))))
 0(0(1(x1))) -> 2(0(2(0(2(1(x1))))))
 0(0(1(x1))) -> 2(0(2(1(0(2(x1))))))
 0(0(1(x1))) -> 2(0(2(1(0(4(x1))))))
 0(0(1(x1))) -> 2(0(2(1(2(0(x1))))))
 0(0(1(x1))) -> 2(0(2(3(1(0(x1))))))
 0(0(1(x1))) -> 2(0(2(5(1(0(x1))))))
 0(0(1(x1))) -> 2(1(0(2(0(2(x1))))))
 0(0(1(x1))) -> 2(1(0(2(2(0(x1))))))
 0(0(1(x1))) -> 2(1(0(2(5(0(x1))))))
 0(0(1(x1))) -> 2(1(2(0(2(0(x1))))))
 0(0(1(x1))) -> 2(1(2(0(4(0(x1))))))
 0(0(1(x1))) -> 2(1(4(0(2(0(x1))))))
 0(0(1(x1))) -> 2(1(4(0(4(0(x1))))))
 0(0(1(x1))) -> 2(2(0(2(1(0(x1))))))
 0(0(1(x1))) -> 2(2(0(4(1(0(x1))))))
 0(0(1(x1))) -> 2(2(1(0(2(0(x1))))))
 0(0(1(x1))) -> 3(1(2(0(2(0(x1))))))
 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1))))))
 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1))))))
 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1))))))
 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1))))))
 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1))))))
 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1))))))
 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1))))))
 0(0(3(1(x1)))) -> 2(0(1(3(0(x1)))))
 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1))))))
 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1))))))
 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1))))))
 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1))))))
 0(0(5(1(x1)))) -> 0(2(5(1(0(x1)))))
 0(0(5(1(x1)))) -> 0(5(0(2(1(x1)))))
 0(0(5(1(x1)))) -> 2(5(0(0(1(x1)))))
 0(0(5(1(x1)))) -> 2(5(1(0(0(x1)))))
 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1))))))
 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1))))))
 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1))))))
 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1))))))
 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1))))))
 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1))))))
 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1))))))
 0(1(0(1(x1)))) -> 0(0(1(2(1(x1)))))
 0(1(0(1(x1)))) -> 1(2(1(0(0(x1)))))
 0(1(0(1(x1)))) -> 2(1(0(0(1(x1)))))
 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1))))))
 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1))))))
 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1))))))
 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1))))))
 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1))))))
 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1))))))
 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1))))))
 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1))))))
 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1))))))
 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1))))))
 0(5(0(1(x1)))) -> 2(5(0(0(1(x1)))))
 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1))))))
 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1))))))
 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1))))))
 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1))))))
 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1))))))
 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1))))))
 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1))))))
 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1))))))
 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1))))))
 0(5(1(1(x1)))) -> 1(2(5(1(0(x1)))))
 0(5(1(1(x1)))) -> 2(5(0(1(1(x1)))))
 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1))))))
 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1))))))
 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1))))))
 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1))))))
 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1))))))
 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1))))))
 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1))))))
 0(5(5(1(x1)))) -> 2(5(5(1(0(x1)))))
 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1))))))
 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1))))))
 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1))))))
 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1))))))
 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1))))))
 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1))))))
 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1))))))
 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1))))))
 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1))))))
 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1))))))
 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1))))))
 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1))))))
 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1))))))
 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1))))))
 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1))))))
 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1))))))
 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1))))))
 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1))))))
 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1))))))
 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1))))))
 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1))))))
 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1))))))
 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1))))))
 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1))))))
 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1))))))
 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1))))))
 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1))))))
 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1))))))
 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1))))))
 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1))))))
 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1))))))
 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1))))))
 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1))))))
 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1))))))
 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1))))))
 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1))))))
 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1))))))
 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1))))))
 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1))))))
 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1))))))
 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1))))))
 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1))))))
 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1))))))
 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1))))))

Proof:
 Bounds Processor:
  bound: 1
  enrichment: match
  automaton:
   final states: {6,5}
   transitions:
    11(25) -> 26*
    11(10) -> 11*
    11(12) -> 13*
    11(7) -> 8*
    11(31) -> 32*
    11(23) -> 24*
    21(11) -> 12*
    41(47) -> 48*
    41(39) -> 40*
    41(9) -> 10*
    41(41) -> 42*
    41(33) -> 34*
    01(8) -> 9*
    00(2) -> 5*
    00(4) -> 5*
    00(1) -> 5*
    00(3) -> 5*
    10(2) -> 1*
    10(4) -> 1*
    10(1) -> 1*
    10(3) -> 1*
    20(2) -> 2*
    20(4) -> 2*
    20(1) -> 2*
    20(3) -> 2*
    30(2) -> 3*
    30(4) -> 3*
    30(1) -> 3*
    30(3) -> 3*
    40(2) -> 4*
    40(4) -> 4*
    40(1) -> 4*
    40(3) -> 4*
    50(2) -> 6*
    50(4) -> 6*
    50(1) -> 6*
    50(3) -> 6*
    1 -> 41,25
    2 -> 33,7
    3 -> 47,31
    4 -> 39,23
    13 -> 9,5
    24 -> 8*
    26 -> 8*
    32 -> 8*
    34 -> 8*
    40 -> 8*
    42 -> 8*
    48 -> 8*
  problem:
   
  Qed

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputICFP 2010 58301

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
YES(?,O(n^1))
InputICFP 2010 58301

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  0(0(1(x1))) -> 1(2(0(2(0(x1)))))
     , 0(0(1(x1))) -> 2(0(2(1(0(x1)))))
     , 0(0(1(x1))) -> 2(1(0(2(0(x1)))))
     , 0(0(1(x1))) -> 1(1(2(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(0(2(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(0(3(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(0(4(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(3(0(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(5(0(x1))))))
     , 0(0(1(x1))) -> 1(2(0(3(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(1(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(2(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(2(0(3(0(x1))))))
     , 0(0(1(x1))) -> 1(2(2(0(4(0(x1))))))
     , 0(0(1(x1))) -> 1(2(3(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(5(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(3(2(0(2(0(x1))))))
     , 0(0(1(x1))) -> 2(0(1(2(0(2(x1))))))
     , 0(0(1(x1))) -> 2(0(1(2(2(0(x1))))))
     , 0(0(1(x1))) -> 2(0(1(3(0(2(x1))))))
     , 0(0(1(x1))) -> 2(0(2(0(2(1(x1))))))
     , 0(0(1(x1))) -> 2(0(2(1(0(2(x1))))))
     , 0(0(1(x1))) -> 2(0(2(1(0(4(x1))))))
     , 0(0(1(x1))) -> 2(0(2(1(2(0(x1))))))
     , 0(0(1(x1))) -> 2(0(2(3(1(0(x1))))))
     , 0(0(1(x1))) -> 2(0(2(5(1(0(x1))))))
     , 0(0(1(x1))) -> 2(1(0(2(0(2(x1))))))
     , 0(0(1(x1))) -> 2(1(0(2(2(0(x1))))))
     , 0(0(1(x1))) -> 2(1(0(2(5(0(x1))))))
     , 0(0(1(x1))) -> 2(1(2(0(2(0(x1))))))
     , 0(0(1(x1))) -> 2(1(2(0(4(0(x1))))))
     , 0(0(1(x1))) -> 2(1(4(0(2(0(x1))))))
     , 0(0(1(x1))) -> 2(1(4(0(4(0(x1))))))
     , 0(0(1(x1))) -> 2(2(0(2(1(0(x1))))))
     , 0(0(1(x1))) -> 2(2(0(4(1(0(x1))))))
     , 0(0(1(x1))) -> 2(2(1(0(2(0(x1))))))
     , 0(0(1(x1))) -> 3(1(2(0(2(0(x1))))))
     , 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1))))))
     , 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1))))))
     , 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1))))))
     , 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1))))))
     , 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1))))))
     , 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1))))))
     , 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1))))))
     , 0(0(3(1(x1)))) -> 2(0(1(3(0(x1)))))
     , 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1))))))
     , 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1))))))
     , 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1))))))
     , 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1))))))
     , 0(0(5(1(x1)))) -> 0(2(5(1(0(x1)))))
     , 0(0(5(1(x1)))) -> 0(5(0(2(1(x1)))))
     , 0(0(5(1(x1)))) -> 2(5(0(0(1(x1)))))
     , 0(0(5(1(x1)))) -> 2(5(1(0(0(x1)))))
     , 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1))))))
     , 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1))))))
     , 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1))))))
     , 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1))))))
     , 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1))))))
     , 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1))))))
     , 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1))))))
     , 0(1(0(1(x1)))) -> 0(0(1(2(1(x1)))))
     , 0(1(0(1(x1)))) -> 1(2(1(0(0(x1)))))
     , 0(1(0(1(x1)))) -> 2(1(0(0(1(x1)))))
     , 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1))))))
     , 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1))))))
     , 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1))))))
     , 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1))))))
     , 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1))))))
     , 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1))))))
     , 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1))))))
     , 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1))))))
     , 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1))))))
     , 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1))))))
     , 0(5(0(1(x1)))) -> 2(5(0(0(1(x1)))))
     , 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1))))))
     , 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1))))))
     , 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1))))))
     , 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1))))))
     , 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1))))))
     , 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1))))))
     , 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1))))))
     , 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1))))))
     , 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1))))))
     , 0(5(1(1(x1)))) -> 1(2(5(1(0(x1)))))
     , 0(5(1(1(x1)))) -> 2(5(0(1(1(x1)))))
     , 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1))))))
     , 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1))))))
     , 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1))))))
     , 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1))))))
     , 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1))))))
     , 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1))))))
     , 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1))))))
     , 0(5(5(1(x1)))) -> 2(5(5(1(0(x1)))))
     , 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1))))))
     , 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1))))))
     , 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1))))))
     , 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1))))))
     , 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1))))))
     , 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1))))))
     , 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1))))))
     , 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1))))))
     , 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1))))))
     , 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1))))))
     , 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1))))))
     , 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1))))))
     , 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1))))))
     , 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1))))))
     , 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1))))))
     , 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1))))))
     , 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1))))))
     , 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1))))))
     , 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1))))))
     , 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1))))))
     , 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1))))))
     , 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1))))))
     , 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1))))))
     , 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1))))))
     , 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1))))))
     , 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1))))))
     , 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1))))))
     , 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
     , 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1))))))
     , 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1))))))
     , 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1))))))
     , 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1))))))
     , 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1))))))
     , 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1))))))
     , 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1))))))
     , 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1))))))
     , 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1))))))
     , 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1))))))
     , 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
     , 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1))))))
     , 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1))))))
     , 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1))))))
     , 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1))))))
     , 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1))))))
     , 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1))))))
     , 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1))))))}

Proof Output:    
  'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:
  
  Details:
  --------
    'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
     'Bounds with minimal-enrichment and initial automaton 'match''
     --------------------------------------------------------------
     Answer:           YES(?,O(n^1))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  0(0(1(x1))) -> 1(2(0(2(0(x1)))))
          , 0(0(1(x1))) -> 2(0(2(1(0(x1)))))
          , 0(0(1(x1))) -> 2(1(0(2(0(x1)))))
          , 0(0(1(x1))) -> 1(1(2(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(0(2(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(0(3(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(0(4(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(3(0(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(5(0(x1))))))
          , 0(0(1(x1))) -> 1(2(0(3(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(1(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(2(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(2(0(3(0(x1))))))
          , 0(0(1(x1))) -> 1(2(2(0(4(0(x1))))))
          , 0(0(1(x1))) -> 1(2(3(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(5(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(3(2(0(2(0(x1))))))
          , 0(0(1(x1))) -> 2(0(1(2(0(2(x1))))))
          , 0(0(1(x1))) -> 2(0(1(2(2(0(x1))))))
          , 0(0(1(x1))) -> 2(0(1(3(0(2(x1))))))
          , 0(0(1(x1))) -> 2(0(2(0(2(1(x1))))))
          , 0(0(1(x1))) -> 2(0(2(1(0(2(x1))))))
          , 0(0(1(x1))) -> 2(0(2(1(0(4(x1))))))
          , 0(0(1(x1))) -> 2(0(2(1(2(0(x1))))))
          , 0(0(1(x1))) -> 2(0(2(3(1(0(x1))))))
          , 0(0(1(x1))) -> 2(0(2(5(1(0(x1))))))
          , 0(0(1(x1))) -> 2(1(0(2(0(2(x1))))))
          , 0(0(1(x1))) -> 2(1(0(2(2(0(x1))))))
          , 0(0(1(x1))) -> 2(1(0(2(5(0(x1))))))
          , 0(0(1(x1))) -> 2(1(2(0(2(0(x1))))))
          , 0(0(1(x1))) -> 2(1(2(0(4(0(x1))))))
          , 0(0(1(x1))) -> 2(1(4(0(2(0(x1))))))
          , 0(0(1(x1))) -> 2(1(4(0(4(0(x1))))))
          , 0(0(1(x1))) -> 2(2(0(2(1(0(x1))))))
          , 0(0(1(x1))) -> 2(2(0(4(1(0(x1))))))
          , 0(0(1(x1))) -> 2(2(1(0(2(0(x1))))))
          , 0(0(1(x1))) -> 3(1(2(0(2(0(x1))))))
          , 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1))))))
          , 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1))))))
          , 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1))))))
          , 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1))))))
          , 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1))))))
          , 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1))))))
          , 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1))))))
          , 0(0(3(1(x1)))) -> 2(0(1(3(0(x1)))))
          , 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1))))))
          , 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1))))))
          , 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1))))))
          , 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1))))))
          , 0(0(5(1(x1)))) -> 0(2(5(1(0(x1)))))
          , 0(0(5(1(x1)))) -> 0(5(0(2(1(x1)))))
          , 0(0(5(1(x1)))) -> 2(5(0(0(1(x1)))))
          , 0(0(5(1(x1)))) -> 2(5(1(0(0(x1)))))
          , 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1))))))
          , 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1))))))
          , 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1))))))
          , 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1))))))
          , 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1))))))
          , 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1))))))
          , 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1))))))
          , 0(1(0(1(x1)))) -> 0(0(1(2(1(x1)))))
          , 0(1(0(1(x1)))) -> 1(2(1(0(0(x1)))))
          , 0(1(0(1(x1)))) -> 2(1(0(0(1(x1)))))
          , 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1))))))
          , 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1))))))
          , 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1))))))
          , 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1))))))
          , 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1))))))
          , 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1))))))
          , 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1))))))
          , 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1))))))
          , 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1))))))
          , 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1))))))
          , 0(5(0(1(x1)))) -> 2(5(0(0(1(x1)))))
          , 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1))))))
          , 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1))))))
          , 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1))))))
          , 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1))))))
          , 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1))))))
          , 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1))))))
          , 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1))))))
          , 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1))))))
          , 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1))))))
          , 0(5(1(1(x1)))) -> 1(2(5(1(0(x1)))))
          , 0(5(1(1(x1)))) -> 2(5(0(1(1(x1)))))
          , 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1))))))
          , 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1))))))
          , 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1))))))
          , 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1))))))
          , 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1))))))
          , 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1))))))
          , 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1))))))
          , 0(5(5(1(x1)))) -> 2(5(5(1(0(x1)))))
          , 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1))))))
          , 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1))))))
          , 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1))))))
          , 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1))))))
          , 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1))))))
          , 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1))))))
          , 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1))))))
          , 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1))))))
          , 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1))))))
          , 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1))))))
          , 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1))))))
          , 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1))))))
          , 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1))))))
          , 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1))))))
          , 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1))))))
          , 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1))))))
          , 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1))))))
          , 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1))))))
          , 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1))))))
          , 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1))))))
          , 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1))))))
          , 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1))))))
          , 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1))))))
          , 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1))))))
          , 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1))))))
          , 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1))))))
          , 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1))))))
          , 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
          , 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1))))))
          , 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1))))))
          , 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1))))))
          , 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1))))))
          , 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1))))))
          , 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1))))))
          , 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1))))))
          , 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1))))))
          , 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1))))))
          , 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1))))))
          , 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
          , 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1))))))
          , 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1))))))
          , 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1))))))
          , 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1))))))
          , 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1))))))
          , 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1))))))
          , 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1))))))}
     
     Proof Output:    
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  0_0(2) -> 1
        , 0_1(7) -> 6
        , 1_0(2) -> 2
        , 1_1(2) -> 7
        , 1_1(3) -> 1
        , 1_1(3) -> 6
        , 1_1(5) -> 4
        , 2_0(2) -> 2
        , 2_1(4) -> 3
        , 3_0(2) -> 2
        , 4_0(2) -> 2
        , 4_1(2) -> 7
        , 4_1(6) -> 5
        , 5_0(2) -> 1}

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputICFP 2010 58301

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
YES(?,O(n^1))
InputICFP 2010 58301

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    runtime-complexity with respect to
  Rules:
    {  0(0(1(x1))) -> 1(2(0(2(0(x1)))))
     , 0(0(1(x1))) -> 2(0(2(1(0(x1)))))
     , 0(0(1(x1))) -> 2(1(0(2(0(x1)))))
     , 0(0(1(x1))) -> 1(1(2(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(0(2(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(0(3(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(0(4(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(3(0(x1))))))
     , 0(0(1(x1))) -> 1(2(0(2(5(0(x1))))))
     , 0(0(1(x1))) -> 1(2(0(3(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(1(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(2(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(2(0(3(0(x1))))))
     , 0(0(1(x1))) -> 1(2(2(0(4(0(x1))))))
     , 0(0(1(x1))) -> 1(2(3(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(2(5(0(2(0(x1))))))
     , 0(0(1(x1))) -> 1(3(2(0(2(0(x1))))))
     , 0(0(1(x1))) -> 2(0(1(2(0(2(x1))))))
     , 0(0(1(x1))) -> 2(0(1(2(2(0(x1))))))
     , 0(0(1(x1))) -> 2(0(1(3(0(2(x1))))))
     , 0(0(1(x1))) -> 2(0(2(0(2(1(x1))))))
     , 0(0(1(x1))) -> 2(0(2(1(0(2(x1))))))
     , 0(0(1(x1))) -> 2(0(2(1(0(4(x1))))))
     , 0(0(1(x1))) -> 2(0(2(1(2(0(x1))))))
     , 0(0(1(x1))) -> 2(0(2(3(1(0(x1))))))
     , 0(0(1(x1))) -> 2(0(2(5(1(0(x1))))))
     , 0(0(1(x1))) -> 2(1(0(2(0(2(x1))))))
     , 0(0(1(x1))) -> 2(1(0(2(2(0(x1))))))
     , 0(0(1(x1))) -> 2(1(0(2(5(0(x1))))))
     , 0(0(1(x1))) -> 2(1(2(0(2(0(x1))))))
     , 0(0(1(x1))) -> 2(1(2(0(4(0(x1))))))
     , 0(0(1(x1))) -> 2(1(4(0(2(0(x1))))))
     , 0(0(1(x1))) -> 2(1(4(0(4(0(x1))))))
     , 0(0(1(x1))) -> 2(2(0(2(1(0(x1))))))
     , 0(0(1(x1))) -> 2(2(0(4(1(0(x1))))))
     , 0(0(1(x1))) -> 2(2(1(0(2(0(x1))))))
     , 0(0(1(x1))) -> 3(1(2(0(2(0(x1))))))
     , 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1))))))
     , 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1))))))
     , 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1))))))
     , 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1))))))
     , 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1))))))
     , 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1))))))
     , 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1))))))
     , 0(0(3(1(x1)))) -> 2(0(1(3(0(x1)))))
     , 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1))))))
     , 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1))))))
     , 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1))))))
     , 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1))))))
     , 0(0(5(1(x1)))) -> 0(2(5(1(0(x1)))))
     , 0(0(5(1(x1)))) -> 0(5(0(2(1(x1)))))
     , 0(0(5(1(x1)))) -> 2(5(0(0(1(x1)))))
     , 0(0(5(1(x1)))) -> 2(5(1(0(0(x1)))))
     , 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1))))))
     , 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1))))))
     , 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1))))))
     , 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1))))))
     , 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1))))))
     , 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1))))))
     , 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1))))))
     , 0(1(0(1(x1)))) -> 0(0(1(2(1(x1)))))
     , 0(1(0(1(x1)))) -> 1(2(1(0(0(x1)))))
     , 0(1(0(1(x1)))) -> 2(1(0(0(1(x1)))))
     , 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1))))))
     , 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1))))))
     , 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1))))))
     , 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1))))))
     , 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1))))))
     , 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1))))))
     , 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1))))))
     , 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1))))))
     , 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1))))))
     , 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1))))))
     , 0(5(0(1(x1)))) -> 2(5(0(0(1(x1)))))
     , 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1))))))
     , 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1))))))
     , 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1))))))
     , 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1))))))
     , 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1))))))
     , 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1))))))
     , 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1))))))
     , 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1))))))
     , 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1))))))
     , 0(5(1(1(x1)))) -> 1(2(5(1(0(x1)))))
     , 0(5(1(1(x1)))) -> 2(5(0(1(1(x1)))))
     , 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1))))))
     , 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1))))))
     , 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1))))))
     , 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1))))))
     , 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1))))))
     , 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1))))))
     , 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1))))))
     , 0(5(5(1(x1)))) -> 2(5(5(1(0(x1)))))
     , 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1))))))
     , 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1))))))
     , 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1))))))
     , 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1))))))
     , 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1))))))
     , 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1))))))
     , 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1))))))
     , 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1))))))
     , 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1))))))
     , 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1))))))
     , 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1))))))
     , 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1))))))
     , 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1))))))
     , 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1))))))
     , 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1))))))
     , 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1))))))
     , 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1))))))
     , 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1))))))
     , 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1))))))
     , 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1))))))
     , 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1))))))
     , 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1))))))
     , 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1))))))
     , 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1))))))
     , 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1))))))
     , 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1))))))
     , 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1))))))
     , 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
     , 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1))))))
     , 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1))))))
     , 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1))))))
     , 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1))))))
     , 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1))))))
     , 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1))))))
     , 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1))))))
     , 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1))))))
     , 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1))))))
     , 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1))))))
     , 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
     , 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1))))))
     , 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1))))))
     , 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1))))))
     , 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1))))))
     , 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1))))))
     , 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1))))))
     , 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1))))))}

Proof Output:    
  'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:
  
  Details:
  --------
    'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
     'Bounds with minimal-enrichment and initial automaton 'match''
     --------------------------------------------------------------
     Answer:           YES(?,O(n^1))
     Input Problem:    runtime-complexity with respect to
       Rules:
         {  0(0(1(x1))) -> 1(2(0(2(0(x1)))))
          , 0(0(1(x1))) -> 2(0(2(1(0(x1)))))
          , 0(0(1(x1))) -> 2(1(0(2(0(x1)))))
          , 0(0(1(x1))) -> 1(1(2(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(0(2(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(0(3(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(0(4(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(3(0(x1))))))
          , 0(0(1(x1))) -> 1(2(0(2(5(0(x1))))))
          , 0(0(1(x1))) -> 1(2(0(3(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(1(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(2(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(2(0(3(0(x1))))))
          , 0(0(1(x1))) -> 1(2(2(0(4(0(x1))))))
          , 0(0(1(x1))) -> 1(2(3(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(2(5(0(2(0(x1))))))
          , 0(0(1(x1))) -> 1(3(2(0(2(0(x1))))))
          , 0(0(1(x1))) -> 2(0(1(2(0(2(x1))))))
          , 0(0(1(x1))) -> 2(0(1(2(2(0(x1))))))
          , 0(0(1(x1))) -> 2(0(1(3(0(2(x1))))))
          , 0(0(1(x1))) -> 2(0(2(0(2(1(x1))))))
          , 0(0(1(x1))) -> 2(0(2(1(0(2(x1))))))
          , 0(0(1(x1))) -> 2(0(2(1(0(4(x1))))))
          , 0(0(1(x1))) -> 2(0(2(1(2(0(x1))))))
          , 0(0(1(x1))) -> 2(0(2(3(1(0(x1))))))
          , 0(0(1(x1))) -> 2(0(2(5(1(0(x1))))))
          , 0(0(1(x1))) -> 2(1(0(2(0(2(x1))))))
          , 0(0(1(x1))) -> 2(1(0(2(2(0(x1))))))
          , 0(0(1(x1))) -> 2(1(0(2(5(0(x1))))))
          , 0(0(1(x1))) -> 2(1(2(0(2(0(x1))))))
          , 0(0(1(x1))) -> 2(1(2(0(4(0(x1))))))
          , 0(0(1(x1))) -> 2(1(4(0(2(0(x1))))))
          , 0(0(1(x1))) -> 2(1(4(0(4(0(x1))))))
          , 0(0(1(x1))) -> 2(2(0(2(1(0(x1))))))
          , 0(0(1(x1))) -> 2(2(0(4(1(0(x1))))))
          , 0(0(1(x1))) -> 2(2(1(0(2(0(x1))))))
          , 0(0(1(x1))) -> 3(1(2(0(2(0(x1))))))
          , 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1))))))
          , 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1))))))
          , 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1))))))
          , 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1))))))
          , 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1))))))
          , 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1))))))
          , 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1))))))
          , 0(0(3(1(x1)))) -> 2(0(1(3(0(x1)))))
          , 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1))))))
          , 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1))))))
          , 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1))))))
          , 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1))))))
          , 0(0(5(1(x1)))) -> 0(2(5(1(0(x1)))))
          , 0(0(5(1(x1)))) -> 0(5(0(2(1(x1)))))
          , 0(0(5(1(x1)))) -> 2(5(0(0(1(x1)))))
          , 0(0(5(1(x1)))) -> 2(5(1(0(0(x1)))))
          , 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1))))))
          , 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1))))))
          , 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1))))))
          , 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1))))))
          , 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1))))))
          , 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1))))))
          , 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1))))))
          , 0(1(0(1(x1)))) -> 0(0(1(2(1(x1)))))
          , 0(1(0(1(x1)))) -> 1(2(1(0(0(x1)))))
          , 0(1(0(1(x1)))) -> 2(1(0(0(1(x1)))))
          , 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1))))))
          , 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1))))))
          , 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1))))))
          , 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1))))))
          , 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1))))))
          , 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1))))))
          , 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1))))))
          , 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1))))))
          , 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1))))))
          , 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1))))))
          , 0(5(0(1(x1)))) -> 2(5(0(0(1(x1)))))
          , 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1))))))
          , 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1))))))
          , 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1))))))
          , 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1))))))
          , 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1))))))
          , 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1))))))
          , 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1))))))
          , 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1))))))
          , 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1))))))
          , 0(5(1(1(x1)))) -> 1(2(5(1(0(x1)))))
          , 0(5(1(1(x1)))) -> 2(5(0(1(1(x1)))))
          , 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1))))))
          , 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1))))))
          , 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1))))))
          , 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1))))))
          , 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1))))))
          , 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1))))))
          , 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1))))))
          , 0(5(5(1(x1)))) -> 2(5(5(1(0(x1)))))
          , 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1))))))
          , 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1))))))
          , 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1))))))
          , 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1))))))
          , 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1))))))
          , 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1))))))
          , 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1))))))
          , 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1))))))
          , 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1))))))
          , 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1))))))
          , 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1))))))
          , 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1))))))
          , 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1))))))
          , 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1))))))
          , 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1))))))
          , 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1))))))
          , 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1))))))
          , 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1))))))
          , 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1))))))
          , 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1))))))
          , 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1))))))
          , 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1))))))
          , 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1))))))
          , 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1))))))
          , 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1))))))
          , 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1))))))
          , 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1))))))
          , 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
          , 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1))))))
          , 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1))))))
          , 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1))))))
          , 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1))))))
          , 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1))))))
          , 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1))))))
          , 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1))))))
          , 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1))))))
          , 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1))))))
          , 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1))))))
          , 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1))))))
          , 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1))))))
          , 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1))))))
          , 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1))))))
          , 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1))))))
          , 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1))))))
          , 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1))))))
          , 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1))))))}
     
     Proof Output:    
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  0_0(2) -> 1
        , 0_1(7) -> 6
        , 1_0(2) -> 2
        , 1_1(2) -> 7
        , 1_1(3) -> 1
        , 1_1(3) -> 6
        , 1_1(5) -> 4
        , 2_0(2) -> 2
        , 2_1(4) -> 3
        , 3_0(2) -> 2
        , 4_0(2) -> 2
        , 4_1(2) -> 7
        , 4_1(6) -> 5
        , 5_0(2) -> 1}