Tool CaT
stdout:
YES(?,O(n^1))
Problem:
i(0(x1)) -> p(s(p(s(0(p(s(p(s(x1)))))))))
i(s(x1)) -> p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))))))))
j(0(x1)) -> p(s(p(p(s(s(0(p(s(p(s(x1)))))))))))
j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1)))))))))))))
p(p(s(x1))) -> p(x1)
p(s(x1)) -> x1
p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1)))))))))
Proof:
Bounds Processor:
bound: 2
enrichment: match
automaton:
final states: {5,4,3}
transitions:
01(20) -> 21*
01(94) -> 95*
s1(45) -> 46*
s1(92) -> 93*
s1(87) -> 88*
s1(82) -> 83*
s1(44) -> 45*
s1(91) -> 92*
s1(86) -> 87*
s1(81) -> 82*
s1(61) -> 62*
s1(56) -> 57*
s1(51) -> 52*
s1(46) -> 47*
s1(26) -> 27*
s1(21) -> 22*
s1(16) -> 17*
s1(93) -> 94*
s1(88) -> 89*
s1(58) -> 59*
s1(53) -> 54*
s1(23) -> 24*
s1(18) -> 19*
s1(90) -> 91*
s1(85) -> 86*
p1(60) -> 61*
p1(50) -> 51*
p1(62) -> 63*
p1(52) -> 53*
p1(47) -> 48*
p1(22) -> 23*
p1(17) -> 18*
p1(84) -> 85*
p1(59) -> 60*
p1(54) -> 55*
p1(49) -> 50*
p1(24) -> 25*
p1(19) -> 20*
p1(83) -> 84*
p1(48) -> 49*
i1(80) -> 81*
j1(55) -> 56*
p2(112) -> 113*
p2(104) -> 105*
p2(116) -> 117*
p2(106) -> 107*
p2(120) -> 121*
i0(2) -> 3*
i0(1) -> 3*
00(2) -> 1*
00(1) -> 1*
p0(2) -> 5*
p0(1) -> 5*
s0(2) -> 2*
s0(1) -> 2*
j0(2) -> 4*
j0(1) -> 4*
1 -> 5,26
2 -> 5,16
16 -> 121,18
17 -> 44*
18 -> 20,80
20 -> 80*
21 -> 113,61,23
22 -> 58*
23 -> 25,3
25 -> 81,3
26 -> 121,18
27 -> 17*
44 -> 117,120
45 -> 107,49,116
46 -> 48,106
47 -> 90*
51 -> 53*
53 -> 55*
57 -> 21*
58 -> 60,112
61 -> 63,4
63 -> 56,4
81 -> 105*
82 -> 84,104
89 -> 56,4
95 -> 5*
105 -> 85*
107 -> 49*
113 -> 61*
117 -> 50*
121 -> 51,53,55
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:
{ i(0(x1)) -> p(s(p(s(0(p(s(p(s(x1)))))))))
, i(s(x1)) ->
p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))))))))
, j(0(x1)) -> p(s(p(p(s(s(0(p(s(p(s(x1)))))))))))
, j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1)))))))))))))
, p(p(s(x1))) -> p(x1)
, p(s(x1)) -> x1
, p(0(x1)) -> 0(s(s(s(s(s(s(s(s(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:
{ i(0(x1)) -> p(s(p(s(0(p(s(p(s(x1)))))))))
, i(s(x1)) ->
p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))))))))
, j(0(x1)) -> p(s(p(p(s(s(0(p(s(p(s(x1)))))))))))
, j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1)))))))))))))
, p(p(s(x1))) -> p(x1)
, p(s(x1)) -> x1
, p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1)))))))))}
Proof Output:
The problem is match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ i_0(2) -> 1
, i_1(7) -> 11
, i_1(7) -> 27
, i_1(7) -> 31
, 0_0(2) -> 1
, 0_0(2) -> 2
, 0_0(2) -> 7
, 0_0(2) -> 9
, 0_0(2) -> 12
, 0_0(2) -> 14
, 0_0(2) -> 16
, 0_1(7) -> 1
, 0_1(7) -> 4
, 0_1(7) -> 6
, 0_1(7) -> 11
, 0_1(7) -> 27
, 0_1(7) -> 31
, 0_1(32) -> 1
, p_0(2) -> 1
, p_1(3) -> 1
, p_1(3) -> 11
, p_1(3) -> 27
, p_1(3) -> 31
, p_1(5) -> 1
, p_1(5) -> 4
, p_1(5) -> 11
, p_1(5) -> 27
, p_1(5) -> 31
, p_1(8) -> 7
, p_1(10) -> 7
, p_1(10) -> 9
, p_1(13) -> 12
, p_1(15) -> 12
, p_1(15) -> 14
, p_1(17) -> 12
, p_1(17) -> 14
, p_1(17) -> 16
, p_1(18) -> 17
, p_1(19) -> 18
, p_1(20) -> 19
, p_1(23) -> 5
, p_1(26) -> 11
, p_1(26) -> 27
, p_1(26) -> 31
, p_1(28) -> 11
, p_1(28) -> 27
, p_1(28) -> 31
, p_1(29) -> 28
, p_1(30) -> 11
, p_2(5) -> 1
, p_2(5) -> 4
, p_2(5) -> 11
, p_2(5) -> 27
, p_2(5) -> 31
, p_2(10) -> 12
, p_2(10) -> 14
, p_2(10) -> 16
, p_2(21) -> 18
, p_2(22) -> 17
, p_2(30) -> 11
, p_2(30) -> 27
, p_2(30) -> 31
, s_0(2) -> 1
, s_0(2) -> 2
, s_0(2) -> 7
, s_0(2) -> 9
, s_0(2) -> 12
, s_0(2) -> 14
, s_0(2) -> 16
, s_1(2) -> 10
, s_1(2) -> 17
, s_1(4) -> 3
, s_1(5) -> 23
, s_1(6) -> 5
, s_1(9) -> 8
, s_1(10) -> 18
, s_1(10) -> 22
, s_1(11) -> 1
, s_1(11) -> 4
, s_1(11) -> 6
, s_1(11) -> 11
, s_1(11) -> 27
, s_1(11) -> 31
, s_1(14) -> 13
, s_1(16) -> 15
, s_1(20) -> 35
, s_1(21) -> 20
, s_1(22) -> 19
, s_1(22) -> 21
, s_1(24) -> 1
, s_1(24) -> 11
, s_1(25) -> 24
, s_1(26) -> 25
, s_1(27) -> 26
, s_1(30) -> 29
, s_1(31) -> 28
, s_1(31) -> 30
, s_1(33) -> 32
, s_1(34) -> 33
, s_1(35) -> 34
, j_0(2) -> 1
, j_1(12) -> 11}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:
{ i(0(x1)) -> p(s(p(s(0(p(s(p(s(x1)))))))))
, i(s(x1)) ->
p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))))))))
, j(0(x1)) -> p(s(p(p(s(s(0(p(s(p(s(x1)))))))))))
, j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1)))))))))))))
, p(p(s(x1))) -> p(x1)
, p(s(x1)) -> x1
, p(0(x1)) -> 0(s(s(s(s(s(s(s(s(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:
{ i(0(x1)) -> p(s(p(s(0(p(s(p(s(x1)))))))))
, i(s(x1)) ->
p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))))))))
, j(0(x1)) -> p(s(p(p(s(s(0(p(s(p(s(x1)))))))))))
, j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1)))))))))))))
, p(p(s(x1))) -> p(x1)
, p(s(x1)) -> x1
, p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1)))))))))}
Proof Output:
The problem is match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ i_0(2) -> 1
, i_1(7) -> 11
, i_1(7) -> 27
, i_1(7) -> 31
, 0_0(2) -> 1
, 0_0(2) -> 2
, 0_0(2) -> 7
, 0_0(2) -> 9
, 0_0(2) -> 12
, 0_0(2) -> 14
, 0_0(2) -> 16
, 0_1(7) -> 1
, 0_1(7) -> 4
, 0_1(7) -> 6
, 0_1(7) -> 11
, 0_1(7) -> 27
, 0_1(7) -> 31
, 0_1(32) -> 1
, p_0(2) -> 1
, p_1(3) -> 1
, p_1(3) -> 11
, p_1(3) -> 27
, p_1(3) -> 31
, p_1(5) -> 1
, p_1(5) -> 4
, p_1(5) -> 11
, p_1(5) -> 27
, p_1(5) -> 31
, p_1(8) -> 7
, p_1(10) -> 7
, p_1(10) -> 9
, p_1(13) -> 12
, p_1(15) -> 12
, p_1(15) -> 14
, p_1(17) -> 12
, p_1(17) -> 14
, p_1(17) -> 16
, p_1(18) -> 17
, p_1(19) -> 18
, p_1(20) -> 19
, p_1(23) -> 5
, p_1(26) -> 11
, p_1(26) -> 27
, p_1(26) -> 31
, p_1(28) -> 11
, p_1(28) -> 27
, p_1(28) -> 31
, p_1(29) -> 28
, p_1(30) -> 11
, p_2(5) -> 1
, p_2(5) -> 4
, p_2(5) -> 11
, p_2(5) -> 27
, p_2(5) -> 31
, p_2(10) -> 12
, p_2(10) -> 14
, p_2(10) -> 16
, p_2(21) -> 18
, p_2(22) -> 17
, p_2(30) -> 11
, p_2(30) -> 27
, p_2(30) -> 31
, s_0(2) -> 1
, s_0(2) -> 2
, s_0(2) -> 7
, s_0(2) -> 9
, s_0(2) -> 12
, s_0(2) -> 14
, s_0(2) -> 16
, s_1(2) -> 10
, s_1(2) -> 17
, s_1(4) -> 3
, s_1(5) -> 23
, s_1(6) -> 5
, s_1(9) -> 8
, s_1(10) -> 18
, s_1(10) -> 22
, s_1(11) -> 1
, s_1(11) -> 4
, s_1(11) -> 6
, s_1(11) -> 11
, s_1(11) -> 27
, s_1(11) -> 31
, s_1(14) -> 13
, s_1(16) -> 15
, s_1(20) -> 35
, s_1(21) -> 20
, s_1(22) -> 19
, s_1(22) -> 21
, s_1(24) -> 1
, s_1(24) -> 11
, s_1(25) -> 24
, s_1(26) -> 25
, s_1(27) -> 26
, s_1(30) -> 29
, s_1(31) -> 28
, s_1(31) -> 30
, s_1(33) -> 32
, s_1(34) -> 33
, s_1(35) -> 34
, j_0(2) -> 1
, j_1(12) -> 11}