Tool CaT
stdout:
YES(?,O(n^1))
Problem:
0(0(1(x1))) -> 0(0(2(1(x1))))
0(1(3(x1))) -> 0(3(4(1(x1))))
3(1(1(x1))) -> 3(1(4(1(x1))))
3(1(1(x1))) -> 3(4(1(4(4(1(x1))))))
3(1(1(x1))) -> 3(4(4(1(2(4(1(x1)))))))
3(3(1(x1))) -> 3(2(1(3(x1))))
5(0(4(x1))) -> 4(0(5(4(x1))))
0(0(0(1(x1)))) -> 0(0(0(2(1(x1)))))
0(0(1(5(x1)))) -> 0(0(5(1(2(4(x1))))))
0(0(5(3(x1)))) -> 0(0(5(4(3(x1)))))
0(4(1(1(x1)))) -> 0(5(4(1(2(1(x1))))))
0(4(3(1(x1)))) -> 0(3(5(4(1(x1)))))
0(4(3(4(x1)))) -> 0(3(2(4(4(x1)))))
3(0(1(1(x1)))) -> 0(2(1(3(1(x1)))))
3(1(1(5(x1)))) -> 3(1(2(1(5(x1)))))
5(0(1(5(x1)))) -> 0(5(4(1(5(x1)))))
5(2(3(1(x1)))) -> 4(5(2(1(3(x1)))))
5(2(3(5(x1)))) -> 2(4(5(3(5(x1)))))
5(2(5(4(x1)))) -> 2(4(5(4(5(x1)))))
0(1(2(2(1(x1))))) -> 2(4(0(1(2(1(x1))))))
0(1(5(0(4(x1))))) -> 0(2(1(5(4(0(x1))))))
0(1(5(5(3(x1))))) -> 0(5(4(5(1(3(x1))))))
0(2(2(1(5(x1))))) -> 0(2(1(5(4(2(4(4(2(x1)))))))))
0(2(2(3(4(x1))))) -> 0(3(4(4(2(4(4(2(x1))))))))
0(2(5(1(1(x1))))) -> 5(0(1(2(4(1(x1))))))
0(3(3(2(3(x1))))) -> 0(3(3(2(4(3(x1))))))
0(4(4(3(4(x1))))) -> 4(4(4(4(0(3(x1))))))
3(0(1(4(3(x1))))) -> 3(2(0(3(4(1(x1))))))
3(0(4(5(1(x1))))) -> 3(5(4(4(1(0(x1))))))
3(1(1(1(2(x1))))) -> 3(2(1(4(1(2(1(2(x1))))))))
5(2(3(1(4(x1))))) -> 5(2(1(2(1(3(4(x1)))))))
5(3(0(0(4(x1))))) -> 0(0(3(5(4(4(x1))))))
0(1(3(4(3(4(x1)))))) -> 0(3(4(3(4(1(2(x1)))))))
0(1(5(0(4(2(x1)))))) -> 0(5(2(4(1(0(2(x1)))))))
0(2(2(2(3(1(x1)))))) -> 0(2(2(4(2(3(1(x1)))))))
0(4(1(1(0(4(x1)))))) -> 0(4(4(1(4(1(0(x1)))))))
0(4(3(2(2(3(x1)))))) -> 0(2(3(2(4(2(3(x1)))))))
3(0(4(1(2(2(x1)))))) -> 3(2(1(0(4(4(2(x1)))))))
3(1(0(3(5(4(x1)))))) -> 3(0(3(2(1(5(4(x1)))))))
3(2(2(1(1(3(x1)))))) -> 1(2(1(3(2(4(3(x1)))))))
3(3(3(1(3(5(x1)))))) -> 3(2(1(3(3(3(5(x1)))))))
3(4(3(0(4(5(x1)))))) -> 3(5(0(3(4(4(4(x1)))))))
3(4(3(1(4(2(x1)))))) -> 3(4(2(1(3(2(2(4(2(4(x1))))))))))
3(5(0(2(1(4(x1)))))) -> 3(2(4(0(5(1(2(x1)))))))
3(5(5(2(3(1(x1)))))) -> 5(5(4(3(3(2(1(x1)))))))
5(0(1(2(3(1(x1)))))) -> 2(1(0(3(4(5(4(4(4(1(x1))))))))))
5(0(4(3(3(1(x1)))))) -> 0(3(5(1(5(4(4(3(x1))))))))
0(0(1(1(3(3(3(x1))))))) -> 0(3(0(1(3(3(4(1(x1))))))))
0(0(2(5(1(4(2(x1))))))) -> 0(2(1(0(5(4(2(2(x1))))))))
0(2(5(4(4(1(5(x1))))))) -> 1(2(4(5(0(5(4(4(x1))))))))
0(4(2(3(4(5(4(x1))))))) -> 3(0(5(4(4(2(4(0(x1))))))))
0(5(2(1(2(1(4(x1))))))) -> 2(1(2(1(0(5(2(4(x1))))))))
3(2(5(0(4(0(1(x1))))))) -> 0(0(5(4(2(2(4(4(1(3(x1))))))))))
3(4(3(4(0(1(4(x1))))))) -> 3(4(2(4(4(4(0(5(1(2(3(x1)))))))))))
3(5(2(1(0(2(2(x1))))))) -> 2(1(3(0(5(2(2(4(x1))))))))
3(5(2(2(0(1(1(x1))))))) -> 3(2(4(5(0(2(1(2(1(x1)))))))))
3(5(3(4(0(1(4(x1))))))) -> 2(1(5(4(3(4(0(3(x1))))))))
5(1(5(3(1(1(3(x1))))))) -> 2(1(5(1(5(1(3(3(x1))))))))
5(2(0(0(1(0(3(x1))))))) -> 0(5(3(4(2(1(0(0(x1))))))))
5(2(0(4(0(4(3(x1))))))) -> 2(0(5(1(0(4(4(3(x1))))))))
5(2(3(0(0(4(3(x1))))))) -> 0(5(4(2(4(3(0(3(3(x1)))))))))
5(2(3(3(2(0(5(x1))))))) -> 2(0(3(3(5(2(4(5(x1))))))))
5(5(5(0(1(4(3(x1))))))) -> 0(5(4(4(3(4(5(5(1(x1)))))))))
0(2(0(5(5(2(2(3(x1)))))))) -> 5(2(2(4(5(3(2(0(0(x1)))))))))
0(2(5(2(4(0(1(4(x1)))))))) -> 0(3(0(5(2(1(2(2(4(4(x1))))))))))
0(4(1(5(3(2(3(3(x1)))))))) -> 0(5(4(1(2(3(3(4(3(x1)))))))))
0(4(5(0(1(5(2(3(x1)))))))) -> 2(1(2(4(2(5(3(0(5(0(x1))))))))))
0(5(5(1(4(2(0(3(x1)))))))) -> 0(3(5(1(2(0(5(1(2(4(x1))))))))))
3(0(1(3(2(0(4(2(x1)))))))) -> 0(4(2(2(2(4(1(0(3(3(4(x1)))))))))))
3(3(1(1(2(5(5(2(x1)))))))) -> 3(2(1(5(1(2(5(3(1(x1)))))))))
3(5(4(2(5(1(0(4(x1)))))))) -> 4(2(4(2(1(3(5(0(5(4(x1))))))))))
5(0(4(3(0(2(2(3(x1)))))))) -> 0(2(4(4(0(3(2(5(3(x1)))))))))
5(2(2(0(4(0(1(4(x1)))))))) -> 4(2(4(0(0(5(1(0(2(x1)))))))))
5(2(2(0(4(1(5(4(x1)))))))) -> 2(5(2(4(5(0(2(1(4(x1)))))))))
5(3(2(2(1(1(0(5(x1)))))))) -> 0(3(2(5(2(1(5(1(2(x1)))))))))
0(0(4(4(5(3(1(1(5(x1))))))))) -> 2(1(5(4(4(4(1(0(0(3(5(x1)))))))))))
0(1(2(5(1(4(0(3(4(x1))))))))) -> 0(2(4(0(3(1(2(1(2(4(5(x1)))))))))))
0(1(3(1(2(0(1(1(5(x1))))))))) -> 0(5(1(0(1(4(1(1(2(3(x1))))))))))
0(2(0(4(1(5(3(0(4(x1))))))))) -> 4(1(0(3(5(0(0(2(2(4(x1))))))))))
0(4(3(3(4(2(5(0(5(x1))))))))) -> 0(5(0(4(3(2(4(5(3(2(x1))))))))))
3(0(4(3(5(4(3(0(1(x1))))))))) -> 0(3(0(3(4(3(2(4(5(1(x1))))))))))
3(1(5(3(1(1(0(3(2(x1))))))))) -> 5(1(0(3(2(1(3(3(2(1(x1))))))))))
3(3(0(0(1(5(4(0(4(x1))))))))) -> 1(0(2(4(4(0(3(5(0(3(x1))))))))))
5(2(0(3(1(2(5(0(4(x1))))))))) -> 5(1(0(3(5(2(4(2(4(0(x1))))))))))
5(2(0(3(2(0(4(4(1(x1))))))))) -> 0(5(3(0(5(2(2(4(2(4(1(x1)))))))))))
5(5(0(1(1(3(2(5(2(x1))))))))) -> 3(5(1(2(2(0(5(1(5(4(x1))))))))))
5(5(3(2(1(0(1(0(4(x1))))))))) -> 0(2(1(3(5(0(5(4(2(4(1(x1)))))))))))
0(1(3(1(4(2(0(4(5(2(x1)))))))))) -> 4(0(1(2(4(0(2(1(5(1(3(x1)))))))))))
0(4(3(1(3(5(2(5(4(4(x1)))))))))) -> 0(5(5(2(4(4(1(5(4(3(3(x1)))))))))))
3(2(0(2(0(5(2(3(4(5(x1)))))))))) -> 3(5(4(4(2(0(3(2(5(2(0(x1)))))))))))
Proof:
Bounds Processor:
bound: 1
enrichment: match
automaton:
final states: {6,5,4}
transitions:
31(15) -> 16*
21(67) -> 68*
21(37) -> 38*
21(79) -> 80*
21(59) -> 60*
21(81) -> 82*
21(61) -> 62*
21(41) -> 42*
21(63) -> 64*
11(62) -> 63*
11(42) -> 43*
11(17) -> 18*
11(12) -> 13*
11(64) -> 65*
11(14) -> 15*
11(66) -> 67*
11(26) -> 27*
11(38) -> 39*
11(23) -> 24*
41(65) -> 66*
41(25) -> 26*
41(27) -> 28*
41(39) -> 40*
41(58) -> 59*
41(43) -> 44*
41(13) -> 14*
01(45) -> 46*
01(57) -> 58*
51(44) -> 45*
00(2) -> 4*
00(1) -> 4*
00(3) -> 4*
10(2) -> 1*
10(1) -> 1*
10(3) -> 1*
20(2) -> 2*
20(1) -> 2*
20(3) -> 2*
30(2) -> 5*
30(1) -> 5*
30(3) -> 5*
40(2) -> 3*
40(1) -> 3*
40(3) -> 3*
50(2) -> 6*
50(1) -> 6*
50(3) -> 6*
1 -> 79,17
2 -> 61,12
3 -> 81,23
13 -> 41*
14 -> 37,25
16 -> 5*
18 -> 13*
24 -> 13*
28 -> 15*
40 -> 27*
43 -> 57*
46 -> 4*
60 -> 4*
68 -> 15*
80 -> 62*
82 -> 62*
problem:
QedTool IRC1
stdout:
MAYBE
Tool IRC2
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))) -> 0(0(2(1(x1))))
, 0(1(3(x1))) -> 0(3(4(1(x1))))
, 3(1(1(x1))) -> 3(1(4(1(x1))))
, 3(1(1(x1))) -> 3(4(1(4(4(1(x1))))))
, 3(1(1(x1))) -> 3(4(4(1(2(4(1(x1)))))))
, 3(3(1(x1))) -> 3(2(1(3(x1))))
, 5(0(4(x1))) -> 4(0(5(4(x1))))
, 0(0(0(1(x1)))) -> 0(0(0(2(1(x1)))))
, 0(0(1(5(x1)))) -> 0(0(5(1(2(4(x1))))))
, 0(0(5(3(x1)))) -> 0(0(5(4(3(x1)))))
, 0(4(1(1(x1)))) -> 0(5(4(1(2(1(x1))))))
, 0(4(3(1(x1)))) -> 0(3(5(4(1(x1)))))
, 0(4(3(4(x1)))) -> 0(3(2(4(4(x1)))))
, 3(0(1(1(x1)))) -> 0(2(1(3(1(x1)))))
, 3(1(1(5(x1)))) -> 3(1(2(1(5(x1)))))
, 5(0(1(5(x1)))) -> 0(5(4(1(5(x1)))))
, 5(2(3(1(x1)))) -> 4(5(2(1(3(x1)))))
, 5(2(3(5(x1)))) -> 2(4(5(3(5(x1)))))
, 5(2(5(4(x1)))) -> 2(4(5(4(5(x1)))))
, 0(1(2(2(1(x1))))) -> 2(4(0(1(2(1(x1))))))
, 0(1(5(0(4(x1))))) -> 0(2(1(5(4(0(x1))))))
, 0(1(5(5(3(x1))))) -> 0(5(4(5(1(3(x1))))))
, 0(2(2(1(5(x1))))) -> 0(2(1(5(4(2(4(4(2(x1)))))))))
, 0(2(2(3(4(x1))))) -> 0(3(4(4(2(4(4(2(x1))))))))
, 0(2(5(1(1(x1))))) -> 5(0(1(2(4(1(x1))))))
, 0(3(3(2(3(x1))))) -> 0(3(3(2(4(3(x1))))))
, 0(4(4(3(4(x1))))) -> 4(4(4(4(0(3(x1))))))
, 3(0(1(4(3(x1))))) -> 3(2(0(3(4(1(x1))))))
, 3(0(4(5(1(x1))))) -> 3(5(4(4(1(0(x1))))))
, 3(1(1(1(2(x1))))) -> 3(2(1(4(1(2(1(2(x1))))))))
, 5(2(3(1(4(x1))))) -> 5(2(1(2(1(3(4(x1)))))))
, 5(3(0(0(4(x1))))) -> 0(0(3(5(4(4(x1))))))
, 0(1(3(4(3(4(x1)))))) -> 0(3(4(3(4(1(2(x1)))))))
, 0(1(5(0(4(2(x1)))))) -> 0(5(2(4(1(0(2(x1)))))))
, 0(2(2(2(3(1(x1)))))) -> 0(2(2(4(2(3(1(x1)))))))
, 0(4(1(1(0(4(x1)))))) -> 0(4(4(1(4(1(0(x1)))))))
, 0(4(3(2(2(3(x1)))))) -> 0(2(3(2(4(2(3(x1)))))))
, 3(0(4(1(2(2(x1)))))) -> 3(2(1(0(4(4(2(x1)))))))
, 3(1(0(3(5(4(x1)))))) -> 3(0(3(2(1(5(4(x1)))))))
, 3(2(2(1(1(3(x1)))))) -> 1(2(1(3(2(4(3(x1)))))))
, 3(3(3(1(3(5(x1)))))) -> 3(2(1(3(3(3(5(x1)))))))
, 3(4(3(0(4(5(x1)))))) -> 3(5(0(3(4(4(4(x1)))))))
, 3(4(3(1(4(2(x1)))))) -> 3(4(2(1(3(2(2(4(2(4(x1))))))))))
, 3(5(0(2(1(4(x1)))))) -> 3(2(4(0(5(1(2(x1)))))))
, 3(5(5(2(3(1(x1)))))) -> 5(5(4(3(3(2(1(x1)))))))
, 5(0(1(2(3(1(x1)))))) -> 2(1(0(3(4(5(4(4(4(1(x1))))))))))
, 5(0(4(3(3(1(x1)))))) -> 0(3(5(1(5(4(4(3(x1))))))))
, 0(0(1(1(3(3(3(x1))))))) -> 0(3(0(1(3(3(4(1(x1))))))))
, 0(0(2(5(1(4(2(x1))))))) -> 0(2(1(0(5(4(2(2(x1))))))))
, 0(2(5(4(4(1(5(x1))))))) -> 1(2(4(5(0(5(4(4(x1))))))))
, 0(4(2(3(4(5(4(x1))))))) -> 3(0(5(4(4(2(4(0(x1))))))))
, 0(5(2(1(2(1(4(x1))))))) -> 2(1(2(1(0(5(2(4(x1))))))))
, 3(2(5(0(4(0(1(x1))))))) -> 0(0(5(4(2(2(4(4(1(3(x1))))))))))
, 3(4(3(4(0(1(4(x1))))))) -> 3(4(2(4(4(4(0(5(1(2(3(x1)))))))))))
, 3(5(2(1(0(2(2(x1))))))) -> 2(1(3(0(5(2(2(4(x1))))))))
, 3(5(2(2(0(1(1(x1))))))) -> 3(2(4(5(0(2(1(2(1(x1)))))))))
, 3(5(3(4(0(1(4(x1))))))) -> 2(1(5(4(3(4(0(3(x1))))))))
, 5(1(5(3(1(1(3(x1))))))) -> 2(1(5(1(5(1(3(3(x1))))))))
, 5(2(0(0(1(0(3(x1))))))) -> 0(5(3(4(2(1(0(0(x1))))))))
, 5(2(0(4(0(4(3(x1))))))) -> 2(0(5(1(0(4(4(3(x1))))))))
, 5(2(3(0(0(4(3(x1))))))) -> 0(5(4(2(4(3(0(3(3(x1)))))))))
, 5(2(3(3(2(0(5(x1))))))) -> 2(0(3(3(5(2(4(5(x1))))))))
, 5(5(5(0(1(4(3(x1))))))) -> 0(5(4(4(3(4(5(5(1(x1)))))))))
, 0(2(0(5(5(2(2(3(x1)))))))) -> 5(2(2(4(5(3(2(0(0(x1)))))))))
, 0(2(5(2(4(0(1(4(x1)))))))) -> 0(3(0(5(2(1(2(2(4(4(x1))))))))))
, 0(4(1(5(3(2(3(3(x1)))))))) -> 0(5(4(1(2(3(3(4(3(x1)))))))))
, 0(4(5(0(1(5(2(3(x1)))))))) -> 2(1(2(4(2(5(3(0(5(0(x1))))))))))
, 0(5(5(1(4(2(0(3(x1)))))))) -> 0(3(5(1(2(0(5(1(2(4(x1))))))))))
, 3(0(1(3(2(0(4(2(x1)))))))) -> 0(4(2(2(2(4(1(0(3(3(4(x1)))))))))))
, 3(3(1(1(2(5(5(2(x1)))))))) -> 3(2(1(5(1(2(5(3(1(x1)))))))))
, 3(5(4(2(5(1(0(4(x1)))))))) -> 4(2(4(2(1(3(5(0(5(4(x1))))))))))
, 5(0(4(3(0(2(2(3(x1)))))))) -> 0(2(4(4(0(3(2(5(3(x1)))))))))
, 5(2(2(0(4(0(1(4(x1)))))))) -> 4(2(4(0(0(5(1(0(2(x1)))))))))
, 5(2(2(0(4(1(5(4(x1)))))))) -> 2(5(2(4(5(0(2(1(4(x1)))))))))
, 5(3(2(2(1(1(0(5(x1)))))))) -> 0(3(2(5(2(1(5(1(2(x1)))))))))
, 0(0(4(4(5(3(1(1(5(x1))))))))) ->
2(1(5(4(4(4(1(0(0(3(5(x1)))))))))))
, 0(1(2(5(1(4(0(3(4(x1))))))))) ->
0(2(4(0(3(1(2(1(2(4(5(x1)))))))))))
, 0(1(3(1(2(0(1(1(5(x1))))))))) -> 0(5(1(0(1(4(1(1(2(3(x1))))))))))
, 0(2(0(4(1(5(3(0(4(x1))))))))) -> 4(1(0(3(5(0(0(2(2(4(x1))))))))))
, 0(4(3(3(4(2(5(0(5(x1))))))))) -> 0(5(0(4(3(2(4(5(3(2(x1))))))))))
, 3(0(4(3(5(4(3(0(1(x1))))))))) -> 0(3(0(3(4(3(2(4(5(1(x1))))))))))
, 3(1(5(3(1(1(0(3(2(x1))))))))) -> 5(1(0(3(2(1(3(3(2(1(x1))))))))))
, 3(3(0(0(1(5(4(0(4(x1))))))))) -> 1(0(2(4(4(0(3(5(0(3(x1))))))))))
, 5(2(0(3(1(2(5(0(4(x1))))))))) -> 5(1(0(3(5(2(4(2(4(0(x1))))))))))
, 5(2(0(3(2(0(4(4(1(x1))))))))) ->
0(5(3(0(5(2(2(4(2(4(1(x1)))))))))))
, 5(5(0(1(1(3(2(5(2(x1))))))))) -> 3(5(1(2(2(0(5(1(5(4(x1))))))))))
, 5(5(3(2(1(0(1(0(4(x1))))))))) ->
0(2(1(3(5(0(5(4(2(4(1(x1)))))))))))
, 0(1(3(1(4(2(0(4(5(2(x1)))))))))) ->
4(0(1(2(4(0(2(1(5(1(3(x1)))))))))))
, 0(4(3(1(3(5(2(5(4(4(x1)))))))))) ->
0(5(5(2(4(4(1(5(4(3(3(x1)))))))))))
, 3(2(0(2(0(5(2(3(4(5(x1)))))))))) ->
3(5(4(4(2(0(3(2(5(2(0(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))) -> 0(0(2(1(x1))))
, 0(1(3(x1))) -> 0(3(4(1(x1))))
, 3(1(1(x1))) -> 3(1(4(1(x1))))
, 3(1(1(x1))) -> 3(4(1(4(4(1(x1))))))
, 3(1(1(x1))) -> 3(4(4(1(2(4(1(x1)))))))
, 3(3(1(x1))) -> 3(2(1(3(x1))))
, 5(0(4(x1))) -> 4(0(5(4(x1))))
, 0(0(0(1(x1)))) -> 0(0(0(2(1(x1)))))
, 0(0(1(5(x1)))) -> 0(0(5(1(2(4(x1))))))
, 0(0(5(3(x1)))) -> 0(0(5(4(3(x1)))))
, 0(4(1(1(x1)))) -> 0(5(4(1(2(1(x1))))))
, 0(4(3(1(x1)))) -> 0(3(5(4(1(x1)))))
, 0(4(3(4(x1)))) -> 0(3(2(4(4(x1)))))
, 3(0(1(1(x1)))) -> 0(2(1(3(1(x1)))))
, 3(1(1(5(x1)))) -> 3(1(2(1(5(x1)))))
, 5(0(1(5(x1)))) -> 0(5(4(1(5(x1)))))
, 5(2(3(1(x1)))) -> 4(5(2(1(3(x1)))))
, 5(2(3(5(x1)))) -> 2(4(5(3(5(x1)))))
, 5(2(5(4(x1)))) -> 2(4(5(4(5(x1)))))
, 0(1(2(2(1(x1))))) -> 2(4(0(1(2(1(x1))))))
, 0(1(5(0(4(x1))))) -> 0(2(1(5(4(0(x1))))))
, 0(1(5(5(3(x1))))) -> 0(5(4(5(1(3(x1))))))
, 0(2(2(1(5(x1))))) -> 0(2(1(5(4(2(4(4(2(x1)))))))))
, 0(2(2(3(4(x1))))) -> 0(3(4(4(2(4(4(2(x1))))))))
, 0(2(5(1(1(x1))))) -> 5(0(1(2(4(1(x1))))))
, 0(3(3(2(3(x1))))) -> 0(3(3(2(4(3(x1))))))
, 0(4(4(3(4(x1))))) -> 4(4(4(4(0(3(x1))))))
, 3(0(1(4(3(x1))))) -> 3(2(0(3(4(1(x1))))))
, 3(0(4(5(1(x1))))) -> 3(5(4(4(1(0(x1))))))
, 3(1(1(1(2(x1))))) -> 3(2(1(4(1(2(1(2(x1))))))))
, 5(2(3(1(4(x1))))) -> 5(2(1(2(1(3(4(x1)))))))
, 5(3(0(0(4(x1))))) -> 0(0(3(5(4(4(x1))))))
, 0(1(3(4(3(4(x1)))))) -> 0(3(4(3(4(1(2(x1)))))))
, 0(1(5(0(4(2(x1)))))) -> 0(5(2(4(1(0(2(x1)))))))
, 0(2(2(2(3(1(x1)))))) -> 0(2(2(4(2(3(1(x1)))))))
, 0(4(1(1(0(4(x1)))))) -> 0(4(4(1(4(1(0(x1)))))))
, 0(4(3(2(2(3(x1)))))) -> 0(2(3(2(4(2(3(x1)))))))
, 3(0(4(1(2(2(x1)))))) -> 3(2(1(0(4(4(2(x1)))))))
, 3(1(0(3(5(4(x1)))))) -> 3(0(3(2(1(5(4(x1)))))))
, 3(2(2(1(1(3(x1)))))) -> 1(2(1(3(2(4(3(x1)))))))
, 3(3(3(1(3(5(x1)))))) -> 3(2(1(3(3(3(5(x1)))))))
, 3(4(3(0(4(5(x1)))))) -> 3(5(0(3(4(4(4(x1)))))))
, 3(4(3(1(4(2(x1)))))) -> 3(4(2(1(3(2(2(4(2(4(x1))))))))))
, 3(5(0(2(1(4(x1)))))) -> 3(2(4(0(5(1(2(x1)))))))
, 3(5(5(2(3(1(x1)))))) -> 5(5(4(3(3(2(1(x1)))))))
, 5(0(1(2(3(1(x1)))))) -> 2(1(0(3(4(5(4(4(4(1(x1))))))))))
, 5(0(4(3(3(1(x1)))))) -> 0(3(5(1(5(4(4(3(x1))))))))
, 0(0(1(1(3(3(3(x1))))))) -> 0(3(0(1(3(3(4(1(x1))))))))
, 0(0(2(5(1(4(2(x1))))))) -> 0(2(1(0(5(4(2(2(x1))))))))
, 0(2(5(4(4(1(5(x1))))))) -> 1(2(4(5(0(5(4(4(x1))))))))
, 0(4(2(3(4(5(4(x1))))))) -> 3(0(5(4(4(2(4(0(x1))))))))
, 0(5(2(1(2(1(4(x1))))))) -> 2(1(2(1(0(5(2(4(x1))))))))
, 3(2(5(0(4(0(1(x1))))))) -> 0(0(5(4(2(2(4(4(1(3(x1))))))))))
, 3(4(3(4(0(1(4(x1))))))) -> 3(4(2(4(4(4(0(5(1(2(3(x1)))))))))))
, 3(5(2(1(0(2(2(x1))))))) -> 2(1(3(0(5(2(2(4(x1))))))))
, 3(5(2(2(0(1(1(x1))))))) -> 3(2(4(5(0(2(1(2(1(x1)))))))))
, 3(5(3(4(0(1(4(x1))))))) -> 2(1(5(4(3(4(0(3(x1))))))))
, 5(1(5(3(1(1(3(x1))))))) -> 2(1(5(1(5(1(3(3(x1))))))))
, 5(2(0(0(1(0(3(x1))))))) -> 0(5(3(4(2(1(0(0(x1))))))))
, 5(2(0(4(0(4(3(x1))))))) -> 2(0(5(1(0(4(4(3(x1))))))))
, 5(2(3(0(0(4(3(x1))))))) -> 0(5(4(2(4(3(0(3(3(x1)))))))))
, 5(2(3(3(2(0(5(x1))))))) -> 2(0(3(3(5(2(4(5(x1))))))))
, 5(5(5(0(1(4(3(x1))))))) -> 0(5(4(4(3(4(5(5(1(x1)))))))))
, 0(2(0(5(5(2(2(3(x1)))))))) -> 5(2(2(4(5(3(2(0(0(x1)))))))))
, 0(2(5(2(4(0(1(4(x1)))))))) -> 0(3(0(5(2(1(2(2(4(4(x1))))))))))
, 0(4(1(5(3(2(3(3(x1)))))))) -> 0(5(4(1(2(3(3(4(3(x1)))))))))
, 0(4(5(0(1(5(2(3(x1)))))))) -> 2(1(2(4(2(5(3(0(5(0(x1))))))))))
, 0(5(5(1(4(2(0(3(x1)))))))) -> 0(3(5(1(2(0(5(1(2(4(x1))))))))))
, 3(0(1(3(2(0(4(2(x1)))))))) -> 0(4(2(2(2(4(1(0(3(3(4(x1)))))))))))
, 3(3(1(1(2(5(5(2(x1)))))))) -> 3(2(1(5(1(2(5(3(1(x1)))))))))
, 3(5(4(2(5(1(0(4(x1)))))))) -> 4(2(4(2(1(3(5(0(5(4(x1))))))))))
, 5(0(4(3(0(2(2(3(x1)))))))) -> 0(2(4(4(0(3(2(5(3(x1)))))))))
, 5(2(2(0(4(0(1(4(x1)))))))) -> 4(2(4(0(0(5(1(0(2(x1)))))))))
, 5(2(2(0(4(1(5(4(x1)))))))) -> 2(5(2(4(5(0(2(1(4(x1)))))))))
, 5(3(2(2(1(1(0(5(x1)))))))) -> 0(3(2(5(2(1(5(1(2(x1)))))))))
, 0(0(4(4(5(3(1(1(5(x1))))))))) ->
2(1(5(4(4(4(1(0(0(3(5(x1)))))))))))
, 0(1(2(5(1(4(0(3(4(x1))))))))) ->
0(2(4(0(3(1(2(1(2(4(5(x1)))))))))))
, 0(1(3(1(2(0(1(1(5(x1))))))))) -> 0(5(1(0(1(4(1(1(2(3(x1))))))))))
, 0(2(0(4(1(5(3(0(4(x1))))))))) -> 4(1(0(3(5(0(0(2(2(4(x1))))))))))
, 0(4(3(3(4(2(5(0(5(x1))))))))) -> 0(5(0(4(3(2(4(5(3(2(x1))))))))))
, 3(0(4(3(5(4(3(0(1(x1))))))))) -> 0(3(0(3(4(3(2(4(5(1(x1))))))))))
, 3(1(5(3(1(1(0(3(2(x1))))))))) -> 5(1(0(3(2(1(3(3(2(1(x1))))))))))
, 3(3(0(0(1(5(4(0(4(x1))))))))) -> 1(0(2(4(4(0(3(5(0(3(x1))))))))))
, 5(2(0(3(1(2(5(0(4(x1))))))))) -> 5(1(0(3(5(2(4(2(4(0(x1))))))))))
, 5(2(0(3(2(0(4(4(1(x1))))))))) ->
0(5(3(0(5(2(2(4(2(4(1(x1)))))))))))
, 5(5(0(1(1(3(2(5(2(x1))))))))) -> 3(5(1(2(2(0(5(1(5(4(x1))))))))))
, 5(5(3(2(1(0(1(0(4(x1))))))))) ->
0(2(1(3(5(0(5(4(2(4(1(x1)))))))))))
, 0(1(3(1(4(2(0(4(5(2(x1)))))))))) ->
4(0(1(2(4(0(2(1(5(1(3(x1)))))))))))
, 0(4(3(1(3(5(2(5(4(4(x1)))))))))) ->
0(5(5(2(4(4(1(5(4(3(3(x1)))))))))))
, 3(2(0(2(0(5(2(3(4(5(x1)))))))))) ->
3(5(4(4(2(0(3(2(5(2(0(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(10) -> 1
, 0_1(12) -> 15
, 1_0(2) -> 2
, 1_1(2) -> 5
, 1_1(4) -> 3
, 1_1(7) -> 6
, 1_1(9) -> 8
, 1_1(13) -> 12
, 1_1(17) -> 16
, 1_1(19) -> 18
, 1_1(21) -> 20
, 2_0(2) -> 2
, 2_1(2) -> 21
, 2_1(4) -> 9
, 2_1(5) -> 13
, 2_1(14) -> 1
, 2_1(16) -> 3
, 2_1(20) -> 19
, 3_0(2) -> 1
, 3_1(3) -> 1
, 4_0(2) -> 2
, 4_1(4) -> 7
, 4_1(5) -> 4
, 4_1(6) -> 3
, 4_1(8) -> 6
, 4_1(12) -> 11
, 4_1(15) -> 14
, 4_1(18) -> 17
, 5_0(2) -> 1
, 5_1(11) -> 10}Tool RC1
stdout:
MAYBE
Tool RC2
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))) -> 0(0(2(1(x1))))
, 0(1(3(x1))) -> 0(3(4(1(x1))))
, 3(1(1(x1))) -> 3(1(4(1(x1))))
, 3(1(1(x1))) -> 3(4(1(4(4(1(x1))))))
, 3(1(1(x1))) -> 3(4(4(1(2(4(1(x1)))))))
, 3(3(1(x1))) -> 3(2(1(3(x1))))
, 5(0(4(x1))) -> 4(0(5(4(x1))))
, 0(0(0(1(x1)))) -> 0(0(0(2(1(x1)))))
, 0(0(1(5(x1)))) -> 0(0(5(1(2(4(x1))))))
, 0(0(5(3(x1)))) -> 0(0(5(4(3(x1)))))
, 0(4(1(1(x1)))) -> 0(5(4(1(2(1(x1))))))
, 0(4(3(1(x1)))) -> 0(3(5(4(1(x1)))))
, 0(4(3(4(x1)))) -> 0(3(2(4(4(x1)))))
, 3(0(1(1(x1)))) -> 0(2(1(3(1(x1)))))
, 3(1(1(5(x1)))) -> 3(1(2(1(5(x1)))))
, 5(0(1(5(x1)))) -> 0(5(4(1(5(x1)))))
, 5(2(3(1(x1)))) -> 4(5(2(1(3(x1)))))
, 5(2(3(5(x1)))) -> 2(4(5(3(5(x1)))))
, 5(2(5(4(x1)))) -> 2(4(5(4(5(x1)))))
, 0(1(2(2(1(x1))))) -> 2(4(0(1(2(1(x1))))))
, 0(1(5(0(4(x1))))) -> 0(2(1(5(4(0(x1))))))
, 0(1(5(5(3(x1))))) -> 0(5(4(5(1(3(x1))))))
, 0(2(2(1(5(x1))))) -> 0(2(1(5(4(2(4(4(2(x1)))))))))
, 0(2(2(3(4(x1))))) -> 0(3(4(4(2(4(4(2(x1))))))))
, 0(2(5(1(1(x1))))) -> 5(0(1(2(4(1(x1))))))
, 0(3(3(2(3(x1))))) -> 0(3(3(2(4(3(x1))))))
, 0(4(4(3(4(x1))))) -> 4(4(4(4(0(3(x1))))))
, 3(0(1(4(3(x1))))) -> 3(2(0(3(4(1(x1))))))
, 3(0(4(5(1(x1))))) -> 3(5(4(4(1(0(x1))))))
, 3(1(1(1(2(x1))))) -> 3(2(1(4(1(2(1(2(x1))))))))
, 5(2(3(1(4(x1))))) -> 5(2(1(2(1(3(4(x1)))))))
, 5(3(0(0(4(x1))))) -> 0(0(3(5(4(4(x1))))))
, 0(1(3(4(3(4(x1)))))) -> 0(3(4(3(4(1(2(x1)))))))
, 0(1(5(0(4(2(x1)))))) -> 0(5(2(4(1(0(2(x1)))))))
, 0(2(2(2(3(1(x1)))))) -> 0(2(2(4(2(3(1(x1)))))))
, 0(4(1(1(0(4(x1)))))) -> 0(4(4(1(4(1(0(x1)))))))
, 0(4(3(2(2(3(x1)))))) -> 0(2(3(2(4(2(3(x1)))))))
, 3(0(4(1(2(2(x1)))))) -> 3(2(1(0(4(4(2(x1)))))))
, 3(1(0(3(5(4(x1)))))) -> 3(0(3(2(1(5(4(x1)))))))
, 3(2(2(1(1(3(x1)))))) -> 1(2(1(3(2(4(3(x1)))))))
, 3(3(3(1(3(5(x1)))))) -> 3(2(1(3(3(3(5(x1)))))))
, 3(4(3(0(4(5(x1)))))) -> 3(5(0(3(4(4(4(x1)))))))
, 3(4(3(1(4(2(x1)))))) -> 3(4(2(1(3(2(2(4(2(4(x1))))))))))
, 3(5(0(2(1(4(x1)))))) -> 3(2(4(0(5(1(2(x1)))))))
, 3(5(5(2(3(1(x1)))))) -> 5(5(4(3(3(2(1(x1)))))))
, 5(0(1(2(3(1(x1)))))) -> 2(1(0(3(4(5(4(4(4(1(x1))))))))))
, 5(0(4(3(3(1(x1)))))) -> 0(3(5(1(5(4(4(3(x1))))))))
, 0(0(1(1(3(3(3(x1))))))) -> 0(3(0(1(3(3(4(1(x1))))))))
, 0(0(2(5(1(4(2(x1))))))) -> 0(2(1(0(5(4(2(2(x1))))))))
, 0(2(5(4(4(1(5(x1))))))) -> 1(2(4(5(0(5(4(4(x1))))))))
, 0(4(2(3(4(5(4(x1))))))) -> 3(0(5(4(4(2(4(0(x1))))))))
, 0(5(2(1(2(1(4(x1))))))) -> 2(1(2(1(0(5(2(4(x1))))))))
, 3(2(5(0(4(0(1(x1))))))) -> 0(0(5(4(2(2(4(4(1(3(x1))))))))))
, 3(4(3(4(0(1(4(x1))))))) -> 3(4(2(4(4(4(0(5(1(2(3(x1)))))))))))
, 3(5(2(1(0(2(2(x1))))))) -> 2(1(3(0(5(2(2(4(x1))))))))
, 3(5(2(2(0(1(1(x1))))))) -> 3(2(4(5(0(2(1(2(1(x1)))))))))
, 3(5(3(4(0(1(4(x1))))))) -> 2(1(5(4(3(4(0(3(x1))))))))
, 5(1(5(3(1(1(3(x1))))))) -> 2(1(5(1(5(1(3(3(x1))))))))
, 5(2(0(0(1(0(3(x1))))))) -> 0(5(3(4(2(1(0(0(x1))))))))
, 5(2(0(4(0(4(3(x1))))))) -> 2(0(5(1(0(4(4(3(x1))))))))
, 5(2(3(0(0(4(3(x1))))))) -> 0(5(4(2(4(3(0(3(3(x1)))))))))
, 5(2(3(3(2(0(5(x1))))))) -> 2(0(3(3(5(2(4(5(x1))))))))
, 5(5(5(0(1(4(3(x1))))))) -> 0(5(4(4(3(4(5(5(1(x1)))))))))
, 0(2(0(5(5(2(2(3(x1)))))))) -> 5(2(2(4(5(3(2(0(0(x1)))))))))
, 0(2(5(2(4(0(1(4(x1)))))))) -> 0(3(0(5(2(1(2(2(4(4(x1))))))))))
, 0(4(1(5(3(2(3(3(x1)))))))) -> 0(5(4(1(2(3(3(4(3(x1)))))))))
, 0(4(5(0(1(5(2(3(x1)))))))) -> 2(1(2(4(2(5(3(0(5(0(x1))))))))))
, 0(5(5(1(4(2(0(3(x1)))))))) -> 0(3(5(1(2(0(5(1(2(4(x1))))))))))
, 3(0(1(3(2(0(4(2(x1)))))))) -> 0(4(2(2(2(4(1(0(3(3(4(x1)))))))))))
, 3(3(1(1(2(5(5(2(x1)))))))) -> 3(2(1(5(1(2(5(3(1(x1)))))))))
, 3(5(4(2(5(1(0(4(x1)))))))) -> 4(2(4(2(1(3(5(0(5(4(x1))))))))))
, 5(0(4(3(0(2(2(3(x1)))))))) -> 0(2(4(4(0(3(2(5(3(x1)))))))))
, 5(2(2(0(4(0(1(4(x1)))))))) -> 4(2(4(0(0(5(1(0(2(x1)))))))))
, 5(2(2(0(4(1(5(4(x1)))))))) -> 2(5(2(4(5(0(2(1(4(x1)))))))))
, 5(3(2(2(1(1(0(5(x1)))))))) -> 0(3(2(5(2(1(5(1(2(x1)))))))))
, 0(0(4(4(5(3(1(1(5(x1))))))))) ->
2(1(5(4(4(4(1(0(0(3(5(x1)))))))))))
, 0(1(2(5(1(4(0(3(4(x1))))))))) ->
0(2(4(0(3(1(2(1(2(4(5(x1)))))))))))
, 0(1(3(1(2(0(1(1(5(x1))))))))) -> 0(5(1(0(1(4(1(1(2(3(x1))))))))))
, 0(2(0(4(1(5(3(0(4(x1))))))))) -> 4(1(0(3(5(0(0(2(2(4(x1))))))))))
, 0(4(3(3(4(2(5(0(5(x1))))))))) -> 0(5(0(4(3(2(4(5(3(2(x1))))))))))
, 3(0(4(3(5(4(3(0(1(x1))))))))) -> 0(3(0(3(4(3(2(4(5(1(x1))))))))))
, 3(1(5(3(1(1(0(3(2(x1))))))))) -> 5(1(0(3(2(1(3(3(2(1(x1))))))))))
, 3(3(0(0(1(5(4(0(4(x1))))))))) -> 1(0(2(4(4(0(3(5(0(3(x1))))))))))
, 5(2(0(3(1(2(5(0(4(x1))))))))) -> 5(1(0(3(5(2(4(2(4(0(x1))))))))))
, 5(2(0(3(2(0(4(4(1(x1))))))))) ->
0(5(3(0(5(2(2(4(2(4(1(x1)))))))))))
, 5(5(0(1(1(3(2(5(2(x1))))))))) -> 3(5(1(2(2(0(5(1(5(4(x1))))))))))
, 5(5(3(2(1(0(1(0(4(x1))))))))) ->
0(2(1(3(5(0(5(4(2(4(1(x1)))))))))))
, 0(1(3(1(4(2(0(4(5(2(x1)))))))))) ->
4(0(1(2(4(0(2(1(5(1(3(x1)))))))))))
, 0(4(3(1(3(5(2(5(4(4(x1)))))))))) ->
0(5(5(2(4(4(1(5(4(3(3(x1)))))))))))
, 3(2(0(2(0(5(2(3(4(5(x1)))))))))) ->
3(5(4(4(2(0(3(2(5(2(0(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))) -> 0(0(2(1(x1))))
, 0(1(3(x1))) -> 0(3(4(1(x1))))
, 3(1(1(x1))) -> 3(1(4(1(x1))))
, 3(1(1(x1))) -> 3(4(1(4(4(1(x1))))))
, 3(1(1(x1))) -> 3(4(4(1(2(4(1(x1)))))))
, 3(3(1(x1))) -> 3(2(1(3(x1))))
, 5(0(4(x1))) -> 4(0(5(4(x1))))
, 0(0(0(1(x1)))) -> 0(0(0(2(1(x1)))))
, 0(0(1(5(x1)))) -> 0(0(5(1(2(4(x1))))))
, 0(0(5(3(x1)))) -> 0(0(5(4(3(x1)))))
, 0(4(1(1(x1)))) -> 0(5(4(1(2(1(x1))))))
, 0(4(3(1(x1)))) -> 0(3(5(4(1(x1)))))
, 0(4(3(4(x1)))) -> 0(3(2(4(4(x1)))))
, 3(0(1(1(x1)))) -> 0(2(1(3(1(x1)))))
, 3(1(1(5(x1)))) -> 3(1(2(1(5(x1)))))
, 5(0(1(5(x1)))) -> 0(5(4(1(5(x1)))))
, 5(2(3(1(x1)))) -> 4(5(2(1(3(x1)))))
, 5(2(3(5(x1)))) -> 2(4(5(3(5(x1)))))
, 5(2(5(4(x1)))) -> 2(4(5(4(5(x1)))))
, 0(1(2(2(1(x1))))) -> 2(4(0(1(2(1(x1))))))
, 0(1(5(0(4(x1))))) -> 0(2(1(5(4(0(x1))))))
, 0(1(5(5(3(x1))))) -> 0(5(4(5(1(3(x1))))))
, 0(2(2(1(5(x1))))) -> 0(2(1(5(4(2(4(4(2(x1)))))))))
, 0(2(2(3(4(x1))))) -> 0(3(4(4(2(4(4(2(x1))))))))
, 0(2(5(1(1(x1))))) -> 5(0(1(2(4(1(x1))))))
, 0(3(3(2(3(x1))))) -> 0(3(3(2(4(3(x1))))))
, 0(4(4(3(4(x1))))) -> 4(4(4(4(0(3(x1))))))
, 3(0(1(4(3(x1))))) -> 3(2(0(3(4(1(x1))))))
, 3(0(4(5(1(x1))))) -> 3(5(4(4(1(0(x1))))))
, 3(1(1(1(2(x1))))) -> 3(2(1(4(1(2(1(2(x1))))))))
, 5(2(3(1(4(x1))))) -> 5(2(1(2(1(3(4(x1)))))))
, 5(3(0(0(4(x1))))) -> 0(0(3(5(4(4(x1))))))
, 0(1(3(4(3(4(x1)))))) -> 0(3(4(3(4(1(2(x1)))))))
, 0(1(5(0(4(2(x1)))))) -> 0(5(2(4(1(0(2(x1)))))))
, 0(2(2(2(3(1(x1)))))) -> 0(2(2(4(2(3(1(x1)))))))
, 0(4(1(1(0(4(x1)))))) -> 0(4(4(1(4(1(0(x1)))))))
, 0(4(3(2(2(3(x1)))))) -> 0(2(3(2(4(2(3(x1)))))))
, 3(0(4(1(2(2(x1)))))) -> 3(2(1(0(4(4(2(x1)))))))
, 3(1(0(3(5(4(x1)))))) -> 3(0(3(2(1(5(4(x1)))))))
, 3(2(2(1(1(3(x1)))))) -> 1(2(1(3(2(4(3(x1)))))))
, 3(3(3(1(3(5(x1)))))) -> 3(2(1(3(3(3(5(x1)))))))
, 3(4(3(0(4(5(x1)))))) -> 3(5(0(3(4(4(4(x1)))))))
, 3(4(3(1(4(2(x1)))))) -> 3(4(2(1(3(2(2(4(2(4(x1))))))))))
, 3(5(0(2(1(4(x1)))))) -> 3(2(4(0(5(1(2(x1)))))))
, 3(5(5(2(3(1(x1)))))) -> 5(5(4(3(3(2(1(x1)))))))
, 5(0(1(2(3(1(x1)))))) -> 2(1(0(3(4(5(4(4(4(1(x1))))))))))
, 5(0(4(3(3(1(x1)))))) -> 0(3(5(1(5(4(4(3(x1))))))))
, 0(0(1(1(3(3(3(x1))))))) -> 0(3(0(1(3(3(4(1(x1))))))))
, 0(0(2(5(1(4(2(x1))))))) -> 0(2(1(0(5(4(2(2(x1))))))))
, 0(2(5(4(4(1(5(x1))))))) -> 1(2(4(5(0(5(4(4(x1))))))))
, 0(4(2(3(4(5(4(x1))))))) -> 3(0(5(4(4(2(4(0(x1))))))))
, 0(5(2(1(2(1(4(x1))))))) -> 2(1(2(1(0(5(2(4(x1))))))))
, 3(2(5(0(4(0(1(x1))))))) -> 0(0(5(4(2(2(4(4(1(3(x1))))))))))
, 3(4(3(4(0(1(4(x1))))))) -> 3(4(2(4(4(4(0(5(1(2(3(x1)))))))))))
, 3(5(2(1(0(2(2(x1))))))) -> 2(1(3(0(5(2(2(4(x1))))))))
, 3(5(2(2(0(1(1(x1))))))) -> 3(2(4(5(0(2(1(2(1(x1)))))))))
, 3(5(3(4(0(1(4(x1))))))) -> 2(1(5(4(3(4(0(3(x1))))))))
, 5(1(5(3(1(1(3(x1))))))) -> 2(1(5(1(5(1(3(3(x1))))))))
, 5(2(0(0(1(0(3(x1))))))) -> 0(5(3(4(2(1(0(0(x1))))))))
, 5(2(0(4(0(4(3(x1))))))) -> 2(0(5(1(0(4(4(3(x1))))))))
, 5(2(3(0(0(4(3(x1))))))) -> 0(5(4(2(4(3(0(3(3(x1)))))))))
, 5(2(3(3(2(0(5(x1))))))) -> 2(0(3(3(5(2(4(5(x1))))))))
, 5(5(5(0(1(4(3(x1))))))) -> 0(5(4(4(3(4(5(5(1(x1)))))))))
, 0(2(0(5(5(2(2(3(x1)))))))) -> 5(2(2(4(5(3(2(0(0(x1)))))))))
, 0(2(5(2(4(0(1(4(x1)))))))) -> 0(3(0(5(2(1(2(2(4(4(x1))))))))))
, 0(4(1(5(3(2(3(3(x1)))))))) -> 0(5(4(1(2(3(3(4(3(x1)))))))))
, 0(4(5(0(1(5(2(3(x1)))))))) -> 2(1(2(4(2(5(3(0(5(0(x1))))))))))
, 0(5(5(1(4(2(0(3(x1)))))))) -> 0(3(5(1(2(0(5(1(2(4(x1))))))))))
, 3(0(1(3(2(0(4(2(x1)))))))) -> 0(4(2(2(2(4(1(0(3(3(4(x1)))))))))))
, 3(3(1(1(2(5(5(2(x1)))))))) -> 3(2(1(5(1(2(5(3(1(x1)))))))))
, 3(5(4(2(5(1(0(4(x1)))))))) -> 4(2(4(2(1(3(5(0(5(4(x1))))))))))
, 5(0(4(3(0(2(2(3(x1)))))))) -> 0(2(4(4(0(3(2(5(3(x1)))))))))
, 5(2(2(0(4(0(1(4(x1)))))))) -> 4(2(4(0(0(5(1(0(2(x1)))))))))
, 5(2(2(0(4(1(5(4(x1)))))))) -> 2(5(2(4(5(0(2(1(4(x1)))))))))
, 5(3(2(2(1(1(0(5(x1)))))))) -> 0(3(2(5(2(1(5(1(2(x1)))))))))
, 0(0(4(4(5(3(1(1(5(x1))))))))) ->
2(1(5(4(4(4(1(0(0(3(5(x1)))))))))))
, 0(1(2(5(1(4(0(3(4(x1))))))))) ->
0(2(4(0(3(1(2(1(2(4(5(x1)))))))))))
, 0(1(3(1(2(0(1(1(5(x1))))))))) -> 0(5(1(0(1(4(1(1(2(3(x1))))))))))
, 0(2(0(4(1(5(3(0(4(x1))))))))) -> 4(1(0(3(5(0(0(2(2(4(x1))))))))))
, 0(4(3(3(4(2(5(0(5(x1))))))))) -> 0(5(0(4(3(2(4(5(3(2(x1))))))))))
, 3(0(4(3(5(4(3(0(1(x1))))))))) -> 0(3(0(3(4(3(2(4(5(1(x1))))))))))
, 3(1(5(3(1(1(0(3(2(x1))))))))) -> 5(1(0(3(2(1(3(3(2(1(x1))))))))))
, 3(3(0(0(1(5(4(0(4(x1))))))))) -> 1(0(2(4(4(0(3(5(0(3(x1))))))))))
, 5(2(0(3(1(2(5(0(4(x1))))))))) -> 5(1(0(3(5(2(4(2(4(0(x1))))))))))
, 5(2(0(3(2(0(4(4(1(x1))))))))) ->
0(5(3(0(5(2(2(4(2(4(1(x1)))))))))))
, 5(5(0(1(1(3(2(5(2(x1))))))))) -> 3(5(1(2(2(0(5(1(5(4(x1))))))))))
, 5(5(3(2(1(0(1(0(4(x1))))))))) ->
0(2(1(3(5(0(5(4(2(4(1(x1)))))))))))
, 0(1(3(1(4(2(0(4(5(2(x1)))))))))) ->
4(0(1(2(4(0(2(1(5(1(3(x1)))))))))))
, 0(4(3(1(3(5(2(5(4(4(x1)))))))))) ->
0(5(5(2(4(4(1(5(4(3(3(x1)))))))))))
, 3(2(0(2(0(5(2(3(4(5(x1)))))))))) ->
3(5(4(4(2(0(3(2(5(2(0(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(10) -> 1
, 0_1(12) -> 15
, 1_0(2) -> 2
, 1_1(2) -> 5
, 1_1(4) -> 3
, 1_1(7) -> 6
, 1_1(9) -> 8
, 1_1(13) -> 12
, 1_1(17) -> 16
, 1_1(19) -> 18
, 1_1(21) -> 20
, 2_0(2) -> 2
, 2_1(2) -> 21
, 2_1(4) -> 9
, 2_1(5) -> 13
, 2_1(14) -> 1
, 2_1(16) -> 3
, 2_1(20) -> 19
, 3_0(2) -> 1
, 3_1(3) -> 1
, 4_0(2) -> 2
, 4_1(4) -> 7
, 4_1(5) -> 4
, 4_1(6) -> 3
, 4_1(8) -> 6
, 4_1(12) -> 11
, 4_1(15) -> 14
, 4_1(18) -> 17
, 5_0(2) -> 1
, 5_1(11) -> 10}