'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 4(x1) -> 9(6(6(x1)))
, 9(x1) -> 6(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ 2^#(7(x1)) -> c_0()
, 2^#(8(1(x1))) -> c_1()
, 2^#(8(x1)) -> c_2(4^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 4^#(7(x1)) -> c_7()
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 6^#(6(x1)) -> c_18()
, 0^#(3(x1)) -> c_19(5^#(3(x1)))}
The usable rules are:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
The estimated dependency graph contains the following edges:
{2^#(8(x1)) -> c_2(4^#(x1))}
==> {4^#(x1) -> c_14(9^#(6(6(x1))))}
{2^#(8(x1)) -> c_2(4^#(x1))}
==> {4^#(7(x1)) -> c_7()}
{2^#(8(x1)) -> c_2(4^#(x1))}
==> {4^#(x1) -> c_4(5^#(2(3(x1))))}
{5^#(9(x1)) -> c_3(0^#(x1))}
==> {0^#(3(x1)) -> c_19(5^#(3(x1)))}
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
==> {6^#(9(x1)) -> c_12(9^#(x1))}
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
==> {6^#(6(x1)) -> c_18()}
{2^#(8(x1)) -> c_6(7^#(x1))}
==> {7^#(0(x1)) -> c_11(9^#(3(x1)))}
{2^#(8(x1)) -> c_6(7^#(x1))}
==> {7^#(2(x1)) -> c_10(4^#(x1))}
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
==> {6^#(2(x1)) -> c_16(7^#(7(x1)))}
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
==> {6^#(9(x1)) -> c_12(9^#(x1))}
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
==> {6^#(6(x1)) -> c_18()}
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
==> {7^#(0(x1)) -> c_11(9^#(3(x1)))}
{7^#(2(x1)) -> c_10(4^#(x1))}
==> {4^#(x1) -> c_14(9^#(6(6(x1))))}
{7^#(2(x1)) -> c_10(4^#(x1))}
==> {4^#(7(x1)) -> c_7()}
{7^#(2(x1)) -> c_10(4^#(x1))}
==> {4^#(x1) -> c_4(5^#(2(3(x1))))}
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
==> {9^#(x1) -> c_15(6^#(7(x1)))}
{6^#(9(x1)) -> c_12(9^#(x1))}
==> {9^#(x1) -> c_15(6^#(7(x1)))}
{6^#(9(x1)) -> c_12(9^#(x1))}
==> {9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
{6^#(9(x1)) -> c_12(9^#(x1))}
==> {9^#(7(x1)) -> c_9(7^#(5(x1)))}
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
==> {5^#(3(x1)) -> c_5(6^#(0(x1)))}
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
==> {5^#(9(x1)) -> c_3(0^#(x1))}
{4^#(x1) -> c_14(9^#(6(6(x1))))}
==> {9^#(x1) -> c_15(6^#(7(x1)))}
{4^#(x1) -> c_14(9^#(6(6(x1))))}
==> {9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
{4^#(x1) -> c_14(9^#(6(6(x1))))}
==> {9^#(7(x1)) -> c_9(7^#(5(x1)))}
{9^#(x1) -> c_15(6^#(7(x1)))}
==> {6^#(6(x1)) -> c_18()}
{9^#(x1) -> c_15(6^#(7(x1)))}
==> {6^#(9(x1)) -> c_12(9^#(x1))}
{6^#(2(x1)) -> c_16(7^#(7(x1)))}
==> {7^#(0(x1)) -> c_11(9^#(3(x1)))}
{2^#(4(x1)) -> c_17(0^#(7(x1)))}
==> {0^#(3(x1)) -> c_19(5^#(3(x1)))}
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
==> {5^#(3(x1)) -> c_5(6^#(0(x1)))}
We consider the following path(s):
1) { 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
The usable rules for this path are the following:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [5]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [8]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [7]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_2(4^#(x1))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_2(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4^#(x1) -> c_14(9^#(6(6(x1))))}
and weakly orienting the rules
{ 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(x1) -> c_14(9^#(6(6(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [5]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [3]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [8]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{ 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [7]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [4]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(8(x1)) -> 4(x1)}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(8(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [2]
4^#(x1) = [1] x1 + [5]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [5]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [13]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [2]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [4]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [4]
5(x1) = [1] x1 + [10]
9(x1) = [1] x1 + [6]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [10]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [10]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))}
and weakly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [4]
8(x1) = [1] x1 + [4]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [14]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [12]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [14]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [2]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [9]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [9]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [9]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [15]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [15]
5^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [6]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [5]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [14]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(3(x1)) -> 5(3(x1))}
and weakly orienting the rules
{ 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(3(x1)) -> 5(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [9]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [9]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [10]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [11]
5^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [14]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [6]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [4]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 9(7(x1)) -> 7(5(x1))}
Weak Rules:
{ 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 9(7(x1)) -> 7(5(x1))}
Weak Rules:
{ 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(3) -> 28
, 7_0(4) -> 28
, 7_0(9) -> 28
, 7_0(30) -> 35
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 5_0(30) -> 33
, 0_0(3) -> 33
, 0_0(4) -> 33
, 0_0(9) -> 33
, 3_0(3) -> 9
, 3_0(3) -> 30
, 3_0(4) -> 9
, 3_0(4) -> 30
, 3_0(9) -> 9
, 3_0(9) -> 30
, 3_0(33) -> 33
, 6_0(3) -> 31
, 6_0(4) -> 31
, 6_0(9) -> 31
, 6_0(31) -> 30
, 6_0(33) -> 33
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, c_2_0(15) -> 11
, 4^#_0(3) -> 15
, 4^#_0(4) -> 15
, 4^#_0(9) -> 15
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18
, c_5_0(32) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 6^#_0(28) -> 27
, 6^#_0(33) -> 32
, 6^#_0(35) -> 34
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, 9^#_0(30) -> 29
, c_14_0(29) -> 15
, c_15_0(27) -> 26
, c_15_0(34) -> 29
, c_19_0(16) -> 18}
2) { 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [4]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [13]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [15]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [3]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [2]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [10]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [7]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [2]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [11]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [9]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [2]
0^#(x1) = [1] x1 + [3]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [7]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(3(x1)) -> 5(3(x1))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(3(x1)) -> 5(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(7(x1)) -> 1(8(x1))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [6]
1(x1) = [1] x1 + [4]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [7]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [12]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [5]
6(x1) = [1] x1 + [4]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [13]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [8]
6^#(x1) = [1] x1 + [9]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [12]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [15]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(x1) -> 5(2(3(x1)))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(x1) -> 5(2(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [4]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [5]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [13]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(x1) -> 9(6(6(x1)))}
and weakly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 2(7(x1)) -> 1(8(x1))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(x1) -> 9(6(6(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [13]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [8]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 4(x1) -> 9(6(6(x1)))
, 4(x1) -> 5(2(3(x1)))
, 2(7(x1)) -> 1(8(x1))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [3]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [2]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [4]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [5]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(0(x1)) -> 9(3(x1))}
and weakly orienting the rules
{ 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 4(x1) -> 5(2(3(x1)))
, 2(7(x1)) -> 1(8(x1))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(0(x1)) -> 9(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [10]
4(x1) = [1] x1 + [11]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [6]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [13]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [5]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 7(0(x1)) -> 9(3(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 4(x1) -> 5(2(3(x1)))
, 2(7(x1)) -> 1(8(x1))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 7(0(x1)) -> 9(3(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 4(x1) -> 5(2(3(x1)))
, 2(7(x1)) -> 1(8(x1))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(2) -> 6
, 1_0(2) -> 2
, 8_0(2) -> 2
, 5_0(2) -> 4
, 0_0(2) -> 4
, 3_0(2) -> 2
, 3_0(4) -> 4
, 6_0(4) -> 4
, 5^#_0(2) -> 1
, 0^#_0(2) -> 1
, c_5_0(3) -> 1
, 6^#_0(2) -> 1
, 6^#_0(4) -> 3
, 6^#_0(6) -> 5
, 7^#_0(2) -> 1
, 9^#_0(2) -> 1
, c_15_0(5) -> 1
, c_19_0(1) -> 1}
3) { 2^#(8(x1)) -> c_6(7^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
The usable rules for this path are the following:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
and weakly orienting the rules
{ 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [8]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [4]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [9]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [9]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [7]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [3]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [5]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [2]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [6]
7(x1) = [1] x1 + [3]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [4]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [13]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_6(7^#(x1))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_6(7^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [15]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [11]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [7]
c_6(x1) = [1] x1 + [3]
7^#(x1) = [1] x1 + [6]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [7]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [2]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [2]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [6]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [2]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [4]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [3]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(3(x1)) -> 5(3(x1))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(3(x1)) -> 5(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [15]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [4]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [12]
3(x1) = [1] x1 + [8]
6(x1) = [1] x1 + [4]
2^#(x1) = [1] x1 + [10]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [6]
c_3(x1) = [1] x1 + [5]
0^#(x1) = [1] x1 + [7]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [4]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 0(3(x1)) -> 5(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 0(3(x1)) -> 5(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(3) -> 28
, 7_0(4) -> 28
, 7_0(9) -> 28
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 5_0(9) -> 30
, 0_0(3) -> 30
, 0_0(4) -> 30
, 0_0(9) -> 30
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 3_0(30) -> 30
, 6_0(30) -> 30
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18
, c_5_0(29) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 6^#_0(28) -> 27
, 6^#_0(30) -> 29
, c_6_0(23) -> 11
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, c_15_0(27) -> 26
, c_19_0(16) -> 18}
4) { 2^#(8(x1)) -> c_6(7^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
The usable rules for this path are the following:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [7]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{6(2(x1)) -> 7(7(x1))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{6(2(x1)) -> 7(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [3]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [2]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [8]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [10]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [7]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [8]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_6(7^#(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_6(7^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [2]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [2]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [2]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [13]
7(x1) = [1] x1 + [3]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [2]
5(x1) = [1] x1 + [2]
9(x1) = [1] x1 + [10]
0(x1) = [1] x1 + [12]
3(x1) = [1] x1 + [5]
6(x1) = [1] x1 + [3]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [3]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [7]
c_12(x1) = [1] x1 + [5]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 4(7(x1)) -> 1(3(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [12]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [4]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [4]
5(x1) = [1] x1 + [4]
9(x1) = [1] x1 + [11]
0(x1) = [1] x1 + [12]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [14]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [4]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [5]
7^#(x1) = [1] x1 + [4]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [4]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [15]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4^#(x1) -> c_14(9^#(6(6(x1))))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(x1) -> c_14(9^#(6(6(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 4^#(x1) -> c_14(9^#(6(6(x1))))
, 4(7(x1)) -> 1(3(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [5]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [11]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [5]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [3]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [3]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 4(7(x1)) -> 1(3(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 4(7(x1)) -> 1(3(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 9^#(x1) -> c_15(6^#(7(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(3) -> 33
, 7_0(4) -> 33
, 7_0(9) -> 33
, 7_0(28) -> 35
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 5_0(28) -> 31
, 0_0(3) -> 31
, 0_0(4) -> 31
, 0_0(9) -> 31
, 3_0(3) -> 9
, 3_0(3) -> 28
, 3_0(4) -> 9
, 3_0(4) -> 28
, 3_0(9) -> 9
, 3_0(9) -> 28
, 3_0(31) -> 31
, 6_0(3) -> 29
, 6_0(4) -> 29
, 6_0(9) -> 29
, 6_0(29) -> 28
, 6_0(31) -> 31
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, 4^#_0(3) -> 15
, 4^#_0(4) -> 15
, 4^#_0(9) -> 15
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18
, c_5_0(30) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 6^#_0(31) -> 30
, 6^#_0(33) -> 32
, 6^#_0(35) -> 34
, c_6_0(23) -> 11
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, 9^#_0(28) -> 27
, c_14_0(27) -> 15
, c_15_0(32) -> 26
, c_15_0(34) -> 27
, c_19_0(16) -> 18}
5) { 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6^#(6(x1)) -> c_18()}
The usable rules for this path are the following:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 6^#(6(x1)) -> c_18()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{6(2(x1)) -> 7(7(x1))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{6(2(x1)) -> 7(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4^#(x1) -> c_14(9^#(6(6(x1))))}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(x1) -> c_14(9^#(6(6(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [15]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_2(4^#(x1))}
and weakly orienting the rules
{ 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_2(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [3]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [3]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{6^#(9(x1)) -> c_12(9^#(x1))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{6^#(9(x1)) -> c_12(9^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [5]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [2]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
and weakly orienting the rules
{ 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [13]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [10]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [8]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [8]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))}
and weakly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [2]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [4]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [5]
3(x1) = [1] x1 + [1]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [3]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [8]
0^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [5]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [8]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [4]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [6]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [2]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(x1) -> 6(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(7(x1)) -> 7(5(x1))
, 9(x1) -> 6(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(x1) -> 6(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [4]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [15]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(x1) -> 6(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [14]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(x1) -> 6(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [2]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(x1) -> 6(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [7]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [4]
4(x1) = [1] x1 + [3]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [7]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [10]
5^#(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [3]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(x1) -> 6(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(x1) -> 6(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(2(x1)) -> 7(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 6^#(6(x1)) -> c_18()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(2) -> 4
, 7_0(8) -> 11
, 1_0(2) -> 2
, 8_0(2) -> 2
, 5_0(8) -> 6
, 0_0(2) -> 6
, 3_0(2) -> 2
, 3_0(2) -> 8
, 3_0(6) -> 6
, 6_0(2) -> 9
, 6_0(6) -> 6
, 6_0(9) -> 8
, 2^#_0(2) -> 1
, c_2_0(1) -> 1
, 4^#_0(2) -> 1
, 5^#_0(2) -> 1
, 0^#_0(2) -> 1
, c_5_0(5) -> 1
, 6^#_0(2) -> 1
, 6^#_0(4) -> 3
, 6^#_0(6) -> 5
, 6^#_0(11) -> 10
, 7^#_0(2) -> 1
, 9^#_0(2) -> 1
, 9^#_0(8) -> 7
, c_14_0(7) -> 1
, c_15_0(3) -> 1
, c_15_0(10) -> 7
, c_18_0() -> 5
, c_19_0(1) -> 1}
6) { 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
The usable rules for this path are the following:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [2]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [15]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
and weakly orienting the rules
{ 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [4]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [15]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [10]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [5]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [15]
7(x1) = [1] x1 + [10]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [2]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [5]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [9]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [13]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [10]
c_9(x1) = [1] x1 + [8]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [3]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(x1) -> 6(7(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(x1) -> 6(7(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [12]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [5]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{6^#(9(x1)) -> c_12(9^#(x1))}
and weakly orienting the rules
{ 9(x1) -> 6(7(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{6^#(9(x1)) -> c_12(9^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [12]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [4]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(4(x1)) -> c_17(0^#(7(x1)))}
and weakly orienting the rules
{ 6^#(9(x1)) -> c_12(9^#(x1))
, 9(x1) -> 6(7(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(4(x1)) -> c_17(0^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [6]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [4]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [2]
2^#(x1) = [1] x1 + [15]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [6]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [5]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [7]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [3]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9(x1) -> 6(7(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [3]
8(x1) = [1] x1 + [6]
4(x1) = [1] x1 + [7]
5(x1) = [1] x1 + [9]
9(x1) = [1] x1 + [10]
0(x1) = [1] x1 + [12]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [12]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [12]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9(x1) -> 6(7(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [14]
7(x1) = [1] x1 + [8]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [10]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [3]
6(x1) = [1] x1 + [2]
2^#(x1) = [1] x1 + [15]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [14]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [4]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [9]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [13]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9(x1) -> 6(7(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9(x1) -> 6(7(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(3) -> 30
, 7_0(4) -> 30
, 7_0(9) -> 30
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 5_0(9) -> 28
, 0_0(3) -> 28
, 0_0(4) -> 28
, 0_0(9) -> 28
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 3_0(28) -> 28
, 6_0(28) -> 28
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18
, c_5_0(27) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 6^#_0(28) -> 27
, 6^#_0(30) -> 29
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, c_15_0(29) -> 26
, c_19_0(16) -> 18}
7) { 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6^#(6(x1)) -> c_18()}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [5]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{6^#(2(x1)) -> c_16(7^#(7(x1)))}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{6^#(2(x1)) -> c_16(7^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [8]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{6^#(9(x1)) -> c_12(9^#(x1))}
and weakly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{6^#(9(x1)) -> c_12(9^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{ 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [3]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [5]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)}
and weakly orienting the rules
{ 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [10]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [11]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [6]
c_7() = [0]
c_8(x1) = [1] x1 + [2]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))}
and weakly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(7(x1)) -> 1(8(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [14]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [2]
5(x1) = [1] x1 + [4]
9(x1) = [1] x1 + [6]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [3]
6(x1) = [1] x1 + [3]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [7]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [7]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [4]
9(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [10]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [13]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [5]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [3]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [7]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [4]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [1] x1 + [4]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(2) -> 4
, 1_0(2) -> 2
, 8_0(2) -> 2
, 5_0(2) -> 6
, 0_0(2) -> 6
, 3_0(2) -> 2
, 3_0(6) -> 6
, 6_0(6) -> 6
, 5^#_0(2) -> 1
, 0^#_0(2) -> 1
, c_5_0(5) -> 1
, 6^#_0(2) -> 1
, 6^#_0(4) -> 3
, 6^#_0(6) -> 5
, 7^#_0(2) -> 1
, 9^#_0(2) -> 1
, c_15_0(3) -> 1
, c_18_0() -> 5
, c_19_0(1) -> 1}
8) { 2^#(8(x1)) -> c_6(7^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6^#(6(x1)) -> c_18()}
The usable rules for this path are the following:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 6^#(6(x1)) -> c_18()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [7]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))}
and weakly orienting the rules
{ 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [1] x1 + [7]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [13]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [13]
c_15(x1) = [1] x1 + [3]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [8]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [4]
c_6(x1) = [1] x1 + [6]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [3]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [8]
6^#(x1) = [1] x1 + [15]
c_6(x1) = [1] x1 + [9]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [15]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [9]
0^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [8]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4^#(x1) -> c_14(9^#(6(6(x1))))}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(x1) -> c_14(9^#(6(6(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [10]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [7]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [9]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [8]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [10]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [3]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [10]
5^#(x1) = [1] x1 + [6]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [6]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [2]
6^#(x1) = [1] x1 + [14]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [1] x1 + [9]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(x1) -> 6(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(x1) -> 6(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [3]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [4]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{ 9(x1) -> 6(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [7]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [4]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9(x1) -> 6(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9(x1) -> 6(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [9]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9(x1) -> 6(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9(x1) -> 6(7(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 6(2(x1)) -> 7(7(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 2(8(1(x1))) -> 8(x1)
, 6^#(6(x1)) -> c_18()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(2) -> 6
, 7_0(8) -> 11
, 1_0(2) -> 2
, 8_0(2) -> 2
, 5_0(2) -> 4
, 0_0(2) -> 4
, 3_0(2) -> 2
, 3_0(2) -> 8
, 3_0(4) -> 4
, 6_0(2) -> 9
, 6_0(4) -> 4
, 6_0(9) -> 8
, 2^#_0(2) -> 1
, 4^#_0(2) -> 1
, 5^#_0(2) -> 1
, 0^#_0(2) -> 1
, c_5_0(3) -> 1
, 6^#_0(2) -> 1
, 6^#_0(4) -> 3
, 6^#_0(6) -> 5
, 6^#_0(11) -> 10
, c_6_0(1) -> 1
, 7^#_0(2) -> 1
, 9^#_0(2) -> 1
, 9^#_0(8) -> 7
, c_14_0(7) -> 1
, c_15_0(5) -> 1
, c_15_0(10) -> 7
, c_18_0() -> 3
, c_19_0(1) -> 1}
9) { 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [9]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [4]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [9]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [9]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [7]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(x1) -> 9(6(6(x1)))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(x1) -> 9(6(6(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [5]
0(x1) = [1] x1 + [4]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [3]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [4]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [5]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [2]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(7(x1)) -> 1(8(x1))}
and weakly orienting the rules
{ 4(x1) -> 9(6(6(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(7(x1)) -> 1(8(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [3]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [13]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [5]
0(x1) = [1] x1 + [4]
3(x1) = [1] x1 + [4]
6(x1) = [1] x1 + [4]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [9]
6^#(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(8(x1)) -> 4(x1)}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(8(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [3]
4(x1) = [1] x1 + [10]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [10]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [11]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [3]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(0(x1)) -> 9(3(x1))}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(0(x1)) -> 9(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [2]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [2]
4(x1) = [1] x1 + [4]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [5]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [3]
c_12(x1) = [1] x1 + [3]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(2(x1)) -> 4(x1)}
and weakly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(2(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [2]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [3]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [2]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [4]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [2]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [2]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ 7_0(3) -> 30
, 7_0(4) -> 30
, 7_0(9) -> 30
, 7_1(3) -> 36
, 7_1(4) -> 36
, 7_1(9) -> 36
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 5_0(9) -> 28
, 5_1(34) -> 31
, 5_1(34) -> 37
, 0_0(3) -> 28
, 0_0(4) -> 28
, 0_0(9) -> 28
, 0_1(3) -> 31
, 0_1(4) -> 31
, 0_1(9) -> 31
, 0_2(3) -> 37
, 0_2(4) -> 37
, 0_2(9) -> 37
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 3_1(3) -> 34
, 3_1(4) -> 34
, 3_1(9) -> 34
, 3_2(37) -> 28
, 3_2(37) -> 31
, 3_2(37) -> 37
, 6_1(31) -> 28
, 6_2(37) -> 31
, 6_2(37) -> 37
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 5^#_1(34) -> 33
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18
, c_5_0(27) -> 16
, c_5_1(32) -> 16
, c_5_2(38) -> 33
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 6^#_0(28) -> 27
, 6^#_0(30) -> 29
, 6^#_1(31) -> 32
, 6^#_1(36) -> 35
, 6^#_2(37) -> 38
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, c_15_0(29) -> 26
, c_15_1(35) -> 26
, c_19_0(16) -> 18
, c_19_1(33) -> 18}
10)
{ 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6^#(6(x1)) -> c_18()}
The usable rules for this path are the following:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 6^#(6(x1)) -> c_18()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [15]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(4(x1)) -> c_17(0^#(7(x1)))}
and weakly orienting the rules
{ 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(4(x1)) -> c_17(0^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
and weakly orienting the rules
{ 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [15]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [5]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [9]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [5]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [4]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [11]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [3]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [14]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [2]
5(x1) = [1] x1 + [15]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [5]
6(x1) = [1] x1 + [3]
2^#(x1) = [1] x1 + [15]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [2]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [5]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [9]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [12]
c_9(x1) = [1] x1 + [9]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [7]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [3]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [8]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [4]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(7(x1)) -> 7(5(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [9]
4(x1) = [1] x1 + [5]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [5]
3(x1) = [1] x1 + [6]
6(x1) = [1] x1 + [3]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [5]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [8]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [3]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(3(x1)) -> 5(3(x1))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(3(x1)) -> 5(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [4]
8(x1) = [1] x1 + [4]
4(x1) = [1] x1 + [12]
5(x1) = [1] x1 + [12]
9(x1) = [1] x1 + [15]
0(x1) = [1] x1 + [13]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [12]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [14]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [6]
c_9(x1) = [1] x1 + [4]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(0(x1)) -> 9(3(x1))}
and weakly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(0(x1)) -> 9(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [11]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [5]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [5]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [14]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [6]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [12]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [12]
7(x1) = [1] x1 + [7]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [10]
9(x1) = [1] x1 + [12]
0(x1) = [1] x1 + [11]
3(x1) = [1] x1 + [6]
6(x1) = [1] x1 + [4]
2^#(x1) = [1] x1 + [13]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [10]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [5]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [10]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [15]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5(3(x1)) -> 6(0(x1))
, 9(x1) -> 6(7(x1))
, 4(7(x1)) -> 1(3(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(3) -> 30
, 7_0(4) -> 30
, 7_0(9) -> 30
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 5_0(9) -> 28
, 0_0(3) -> 28
, 0_0(4) -> 28
, 0_0(9) -> 28
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 3_0(28) -> 28
, 6_0(28) -> 28
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18
, c_5_0(27) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 6^#_0(28) -> 27
, 6^#_0(30) -> 29
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, c_15_0(29) -> 26
, c_18_0() -> 27
, c_19_0(16) -> 18}
11)
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6^#(6(x1)) -> c_18()}
The usable rules for this path are the following:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 4(x1) -> 9(6(6(x1)))
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 6^#(6(x1)) -> c_18()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_6(7^#(x1))}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_6(7^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [7]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
and weakly orienting the rules
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7^#(0(x1)) -> c_11(9^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [7]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [8]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(3(x1)) -> c_5(6^#(0(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(x1) -> c_15(6^#(7(x1)))}
and weakly orienting the rules
{ 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(x1) -> c_15(6^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [8]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [5]
7^#(x1) = [1] x1 + [4]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [3]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [5]
c_13(x1) = [1] x1 + [9]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [13]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)}
and weakly orienting the rules
{ 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [15]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [15]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [5]
c_12(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [10]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [7]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [9]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [2]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [5]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [5]
c_12(x1) = [1] x1 + [5]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [15]
7(x1) = [1] x1 + [4]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [5]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(3(x1)) -> 5(3(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [13]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [3]
5(x1) = [1] x1 + [4]
9(x1) = [1] x1 + [5]
0(x1) = [1] x1 + [5]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [5]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [6]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [2]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 0(3(x1)) -> 5(3(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 0(3(x1)) -> 5(3(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 4(7(x1)) -> 1(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(x1)) -> 7(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 6^#(6(x1)) -> c_18()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 7_0(3) -> 28
, 7_0(4) -> 28
, 7_0(9) -> 28
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 5_0(9) -> 30
, 0_0(3) -> 30
, 0_0(4) -> 30
, 0_0(9) -> 30
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 3_0(30) -> 30
, 6_0(30) -> 30
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18
, c_5_0(29) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 6^#_0(28) -> 27
, 6^#_0(30) -> 29
, c_6_0(23) -> 11
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, c_15_0(27) -> 26
, c_18_0() -> 29
, c_19_0(16) -> 18}
12)
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 6^#(6(x1)) -> c_18()}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [9]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))}
and weakly orienting the rules
{ 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [15]
5(x1) = [1] x1 + [13]
9(x1) = [1] x1 + [6]
0(x1) = [1] x1 + [14]
3(x1) = [1] x1 + [12]
6(x1) = [1] x1 + [13]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [4]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [12]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [8]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [4]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [11]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9(7(x1)) -> 7(5(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9(7(x1)) -> 7(5(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [2]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [10]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [2]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [8]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [9]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [7]
c_9(x1) = [1] x1 + [3]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
and weakly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9^#(7(x1)) -> c_9(7^#(5(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(x1) -> 9(6(6(x1)))}
and weakly orienting the rules
{ 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(x1) -> 9(6(6(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [12]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [7]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [6]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [1]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
and weakly orienting the rules
{ 4(x1) -> 9(6(6(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(3(x1)) -> c_19(5^#(3(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [10]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [4]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [2]
0^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [1] x1 + [4]
9^#(x1) = [1] x1 + [9]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [3]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(8(x1)) -> 4(x1)}
and weakly orienting the rules
{ 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(8(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(7(x1)) -> 1(8(x1))}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(7(x1)) -> 1(8(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(0(x1)) -> 9(3(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(0(x1)) -> 9(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [4]
4(x1) = [1] x1 + [12]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [5]
0^#(x1) = [1] x1 + [14]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [9]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [4]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(2(x1)) -> 4(x1)}
and weakly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(2(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [2]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [2]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [4]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(x1)) -> 4(x1)
, 0^#(3(x1)) -> c_19(5^#(3(x1)))
, 4(x1) -> 9(6(6(x1)))
, 9^#(7(x1)) -> c_9(7^#(5(x1)))
, 4(x1) -> 5(2(3(x1)))
, 9(x1) -> 6(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 4(7(x1)) -> 1(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 6^#(9(x1)) -> c_12(9^#(x1))
, 9^#(x1) -> c_15(6^#(7(x1)))
, 9^#(5(9(x1))) -> c_13(5^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 5^#(3(x1)) -> c_5(6^#(0(x1)))
, 5^#(9(x1)) -> c_3(0^#(x1))
, 7^#(0(x1)) -> c_11(9^#(3(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ 7_0(3) -> 28
, 7_0(4) -> 28
, 7_0(9) -> 28
, 7_1(3) -> 35
, 7_1(4) -> 35
, 7_1(9) -> 35
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 5_0(9) -> 30
, 5_1(33) -> 31
, 5_1(33) -> 37
, 0_0(3) -> 30
, 0_0(4) -> 30
, 0_0(9) -> 30
, 0_1(3) -> 31
, 0_1(4) -> 31
, 0_1(9) -> 31
, 0_2(3) -> 37
, 0_2(4) -> 37
, 0_2(9) -> 37
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 3_1(3) -> 33
, 3_1(4) -> 33
, 3_1(9) -> 33
, 3_2(37) -> 30
, 3_2(37) -> 31
, 3_2(37) -> 37
, 6_1(31) -> 30
, 6_2(37) -> 31
, 6_2(37) -> 37
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 5^#_1(33) -> 32
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18
, c_5_0(29) -> 16
, c_5_1(36) -> 16
, c_5_1(36) -> 32
, c_5_2(38) -> 32
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 6^#_0(28) -> 27
, 6^#_0(30) -> 29
, 6^#_1(31) -> 36
, 6^#_1(35) -> 34
, 6^#_2(37) -> 38
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, c_15_0(27) -> 26
, c_15_1(34) -> 26
, c_18_1() -> 29
, c_18_2() -> 36
, c_18_2() -> 38
, c_19_0(16) -> 18
, c_19_1(32) -> 18}
13)
{ 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))}
The usable rules for this path are the following:
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 4(x1) -> 9(6(6(x1)))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 4(x1) -> 9(6(6(x1)))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [4]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(8(x1)) -> 4(x1)}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(8(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [12]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4^#(x1) -> c_14(9^#(6(6(x1))))}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(x1) -> c_14(9^#(6(6(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [4]
8(x1) = [1] x1 + [5]
4(x1) = [1] x1 + [3]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [4]
2^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [12]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(x1) -> 9(6(6(x1)))}
and weakly orienting the rules
{ 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(x1) -> 9(6(6(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [13]
8(x1) = [1] x1 + [12]
4(x1) = [1] x1 + [12]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [4]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [12]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))}
and weakly orienting the rules
{ 4(x1) -> 9(6(6(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [2]
8(x1) = [1] x1 + [7]
4(x1) = [1] x1 + [7]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [6]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [2]
2^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [6]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [4]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(7(x1)) -> 1(8(x1))}
and weakly orienting the rules
{ 9(x1) -> 6(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(7(x1)) -> 1(8(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [4]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [4]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(2(x1)) -> 4(x1)}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 9(x1) -> 6(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(2(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 9(x1) -> 6(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [5]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [14]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [6]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [11]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [12]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [6]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 9(x1) -> 6(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 7(2(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 9(x1) -> 6(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2^#(8(x1)) -> c_2(4^#(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(4(x1)) -> 0(7(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 3_0(3) -> 9
, 3_0(3) -> 28
, 3_0(4) -> 9
, 3_0(4) -> 28
, 3_0(9) -> 9
, 3_0(9) -> 28
, 6_0(3) -> 29
, 6_0(4) -> 29
, 6_0(9) -> 29
, 6_0(29) -> 28
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, c_2_0(15) -> 11
, 4^#_0(3) -> 15
, 4^#_0(4) -> 15
, 4^#_0(9) -> 15
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, 9^#_0(28) -> 27
, c_14_0(27) -> 15}
14)
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))}
The usable rules for this path are the following:
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 4(x1) -> 9(6(6(x1)))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 5(3(x1)) -> 6(0(x1))
, 4(7(x1)) -> 1(3(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 4(x1) -> 9(6(6(x1)))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [4]
3(x1) = [1] x1 + [1]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)}
and weakly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))}
and weakly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4^#(x1) -> c_14(9^#(6(6(x1))))}
and weakly orienting the rules
{ 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(x1) -> c_14(9^#(6(6(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7^#(2(x1)) -> c_10(4^#(x1))}
and weakly orienting the rules
{ 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7^#(2(x1)) -> c_10(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [13]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [1]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [13]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [3]
2^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [7]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(7(x1)) -> 1(8(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(7(x1)) -> 1(8(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [6]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [1]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [5]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [3]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [1]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 2(4(x1)) -> 0(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [4]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [4]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [2]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [5]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [2]
7^#(x1) = [1] x1 + [3]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [1] x1 + [2]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_14(9^#(6(6(x1))))
, 9(x1) -> 6(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(1(x1))) -> 8(x1)
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 7(0(x1)) -> 9(3(x1))
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 3_0(3) -> 9
, 3_0(3) -> 28
, 3_0(4) -> 9
, 3_0(4) -> 28
, 3_0(9) -> 9
, 3_0(9) -> 28
, 6_0(3) -> 29
, 6_0(4) -> 29
, 6_0(9) -> 29
, 6_0(29) -> 28
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, 4^#_0(3) -> 15
, 4^#_0(4) -> 15
, 4^#_0(9) -> 15
, c_6_0(23) -> 11
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23
, 9^#_0(3) -> 26
, 9^#_0(4) -> 26
, 9^#_0(9) -> 26
, 9^#_0(28) -> 27
, c_14_0(27) -> 15}
15)
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [5]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_6(7^#(x1))}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_6(7^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4^#(x1) -> c_4(5^#(2(3(x1))))}
and weakly orienting the rules
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(x1) -> c_4(5^#(2(3(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7^#(2(x1)) -> c_10(4^#(x1))}
and weakly orienting the rules
{ 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7^#(2(x1)) -> c_10(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [13]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [5]
5^#(x1) = [1] x1 + [2]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [2]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
and weakly orienting the rules
{ 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [14]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [10]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [6]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [3]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [3]
8(x1) = [1] x1 + [12]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [8]
6(x1) = [1] x1 + [4]
2^#(x1) = [1] x1 + [12]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [9]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(7(x1)) -> 1(8(x1))}
and weakly orienting the rules
{ 2(4(x1)) -> 0(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(7(x1)) -> 1(8(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [4]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [12]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [4]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [9]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [2]
4(x1) = [1] x1 + [2]
5(x1) = [1] x1 + [2]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(0(x1)) -> 9(3(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(0(x1)) -> 9(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
7^#(x1) = [1] x1 + [8]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 7(0(x1)) -> 9(3(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 7(0(x1)) -> 9(3(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))
, 2^#(8(x1)) -> c_6(7^#(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 2_0(9) -> 25
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, 4^#_0(3) -> 15
, 4^#_0(4) -> 15
, 4^#_0(9) -> 15
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 5^#_0(25) -> 24
, c_4_0(24) -> 15
, c_6_0(23) -> 11
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23}
16)
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(x1) -> 9(6(6(x1)))}
and weakly orienting the rules
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(x1) -> 9(6(6(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [9]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
and weakly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [12]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [13]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(0(x1)) -> 9(3(x1))}
and weakly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(0(x1)) -> 9(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [12]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [15]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(7(x1)) -> 1(8(x1))}
and weakly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(7(x1)) -> 1(8(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [2]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(8(x1)) -> 4(x1)}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(8(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [3]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [2]
4(x1) = [1] x1 + [4]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [13]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [8]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(2(x1)) -> 4(x1)}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(2(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [10]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [11]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [5]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [6]
4(x1) = [1] x1 + [14]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [6]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [13]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21}
17)
{2^#(4(x1)) -> c_17(0^#(7(x1)))}
The usable rules for this path are the following:
{ 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 2(8(x1)) -> 7(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 2(4(x1)) -> 0(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 2^#(4(x1)) -> c_17(0^#(7(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(4(x1)) -> c_17(0^#(7(x1)))}
and weakly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(4(x1)) -> c_17(0^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(8(x1)) -> 4(x1)}
and weakly orienting the rules
{ 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(8(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(8(x1)) -> 7(x1)}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(8(x1)) -> 7(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [8]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [2]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 2(8(x1)) -> 7(x1)
, 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [15]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [3]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(8(x1)) -> 7(x1)
, 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [4]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [2]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 5(3(x1)) -> 6(0(x1))
, 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(8(x1)) -> 7(x1)
, 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [15]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [1]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(0(x1)) -> 9(3(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 5(3(x1)) -> 6(0(x1))
, 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(8(x1)) -> 7(x1)
, 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(0(x1)) -> 9(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [2]
7(x1) = [1] x1 + [1]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 7(0(x1)) -> 9(3(x1))
, 4(7(x1)) -> 1(3(x1))
, 5(3(x1)) -> 6(0(x1))
, 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(8(x1)) -> 7(x1)
, 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [4]
9(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [4]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(3(x1)) -> 5(3(x1))}
and weakly orienting the rules
{ 9(x1) -> 6(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 4(7(x1)) -> 1(3(x1))
, 5(3(x1)) -> 6(0(x1))
, 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(8(x1)) -> 7(x1)
, 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(3(x1)) -> 5(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [14]
7(x1) = [1] x1 + [8]
1(x1) = [1] x1 + [8]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [4]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [10]
0(x1) = [1] x1 + [9]
3(x1) = [1] x1 + [3]
6(x1) = [1] x1 + [2]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [1] x1 + [1]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 0(3(x1)) -> 5(3(x1))
, 9(x1) -> 6(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 4(7(x1)) -> 1(3(x1))
, 5(3(x1)) -> 6(0(x1))
, 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(8(x1)) -> 7(x1)
, 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 0(3(x1)) -> 5(3(x1))
, 9(x1) -> 6(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 4(7(x1)) -> 1(3(x1))
, 5(3(x1)) -> 6(0(x1))
, 9(7(x1)) -> 7(5(x1))
, 2(7(x1)) -> 1(8(x1))
, 2(4(x1)) -> 0(7(x1))
, 2(8(x1)) -> 7(x1)
, 2(8(x1)) -> 4(x1)
, 2^#(4(x1)) -> c_17(0^#(7(x1)))
, 7(2(x1)) -> 4(x1)
, 9(5(9(x1))) -> 5(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, 0^#_0(3) -> 18
, 0^#_0(4) -> 18
, 0^#_0(9) -> 18}
18)
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [5]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
and weakly orienting the rules
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(3(x1)) -> 5(3(x1))}
and weakly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(3(x1)) -> 5(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [8]
6(x1) = [1] x1 + [5]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(7(x1)) -> 1(8(x1))}
and weakly orienting the rules
{ 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(7(x1)) -> 1(8(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [4]
6(x1) = [1] x1 + [2]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(0(x1)) -> 9(3(x1))}
and weakly orienting the rules
{ 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(0(x1)) -> 9(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [2]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [1]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 7(0(x1)) -> 9(3(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 7(0(x1)) -> 9(3(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(7(x1)) -> 1(8(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 2(8(x1)) -> 4(x1)
, 7(2(x1)) -> 4(x1)
, 9(7(x1)) -> 7(5(x1))
, 6^#(2(x1)) -> c_16(7^#(7(x1)))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(2(x1)) -> 7(7(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21
, 7^#_0(3) -> 23
, 7^#_0(4) -> 23
, 7^#_0(9) -> 23}
19)
{ 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 2(4(x1)) -> 0(7(x1))
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 0(3(x1)) -> 5(3(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [2]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [9]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [2]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [9]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [3]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [9]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [2]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_2(4^#(x1))}
and weakly orienting the rules
{ 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_2(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [8]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [9]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [3]
2^#(x1) = [1] x1 + [13]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [12]
5^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [9]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{9(7(x1)) -> 7(5(x1))}
and weakly orienting the rules
{ 2^#(8(x1)) -> c_2(4^#(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{9(7(x1)) -> 7(5(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [3]
5^#(x1) = [1] x1 + [2]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(4(x1)) -> 0(7(x1))}
and weakly orienting the rules
{ 9(7(x1)) -> 7(5(x1))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(4(x1)) -> 0(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [5]
5^#(x1) = [1] x1 + [3]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))}
and weakly orienting the rules
{ 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [2]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [1]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [3]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [3]
4(x1) = [1] x1 + [2]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [3]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [1] x1 + [14]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [2]
4^#(x1) = [1] x1 + [8]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(x1) -> 5(2(3(x1)))
, 4(x1) -> 9(6(6(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 2(7(x1)) -> 1(8(x1))
, 4(7(x1)) -> 1(3(x1))
, 2(4(x1)) -> 0(7(x1))
, 9(7(x1)) -> 7(5(x1))
, 2^#(8(x1)) -> c_2(4^#(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 5(9(x1)) -> 0(x1)
, 5(2(6(x1))) -> 6(2(4(x1)))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 2(8(x1)) -> 7(x1)
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 0(3(x1)) -> 5(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 4^#(x1) -> c_4(5^#(2(3(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 2_0(9) -> 18
, 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 2^#_0(3) -> 11
, 2^#_0(4) -> 11
, 2^#_0(9) -> 11
, c_2_0(15) -> 11
, 4^#_0(3) -> 15
, 4^#_0(4) -> 15
, 4^#_0(9) -> 15
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 5^#_0(18) -> 17
, c_4_0(17) -> 15}
20)
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
The usable rules for this path are the following:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2(7(x1)) -> 1(8(x1))
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 4(x1)
, 4(x1) -> 5(2(3(x1)))
, 2(8(x1)) -> 7(x1)
, 4(7(x1)) -> 1(3(x1))
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 5(9(x1)) -> 0(x1)
, 5(3(x1)) -> 6(0(x1))
, 5(2(6(x1))) -> 6(2(4(x1)))
, 7(2(x1)) -> 4(x1)
, 7(0(x1)) -> 9(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(2(x1)) -> 7(7(x1))
, 6(6(x1)) -> 3(x1)
, 0(3(x1)) -> 5(3(x1))
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
and weakly orienting the rules
{ 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [1]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [5]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
and weakly orienting the rules
{ 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [9]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [5]
0(x1) = [1] x1 + [5]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [2]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [12]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(9(x1)) -> 0(x1)}
and weakly orienting the rules
{ 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(9(x1)) -> 0(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [1]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [12]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [13]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(7(x1)) -> 1(8(x1))}
and weakly orienting the rules
{ 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(7(x1)) -> 1(8(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [1]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [1] x1 + [4]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [4]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{2(8(x1)) -> 4(x1)}
and weakly orienting the rules
{ 2(7(x1)) -> 1(8(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2(8(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [8]
4(x1) = [1] x1 + [12]
5(x1) = [1] x1 + [0]
9(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [13]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [1]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{7(2(x1)) -> 4(x1)}
and weakly orienting the rules
{ 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7(2(x1)) -> 4(x1)}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [8]
7(x1) = [1] x1 + [2]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [2]
4(x1) = [1] x1 + [9]
5(x1) = [1] x1 + [1]
9(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [8]
3(x1) = [1] x1 + [0]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{5(3(x1)) -> 6(0(x1))}
and weakly orienting the rules
{ 7(2(x1)) -> 4(x1)
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(3(x1)) -> 6(0(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [9]
7(x1) = [1] x1 + [5]
1(x1) = [1] x1 + [6]
8(x1) = [1] x1 + [6]
4(x1) = [1] x1 + [14]
5(x1) = [1] x1 + [3]
9(x1) = [1] x1 + [6]
0(x1) = [1] x1 + [3]
3(x1) = [1] x1 + [2]
6(x1) = [1] x1 + [1]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [15]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {5(2(6(x1))) -> 6(2(4(x1)))}
Weak Rules:
{ 5(3(x1)) -> 6(0(x1))
, 7(2(x1)) -> 4(x1)
, 2(8(x1)) -> 4(x1)
, 2(7(x1)) -> 1(8(x1))
, 5(9(x1)) -> 0(x1)
, 4(x1) -> 5(2(3(x1)))
, 4(7(x1)) -> 1(3(x1))
, 6(9(x1)) -> 9(x1)
, 6(6(x1)) -> 3(x1)
, 9(7(x1)) -> 7(5(x1))
, 9(5(9(x1))) -> 5(7(x1))
, 9(x1) -> 6(7(x1))
, 5^#(2(6(x1))) -> c_8(6^#(2(4(x1))))
, 6^#(6(x1)) -> c_18()
, 2(8(1(x1))) -> 8(x1)
, 2(8(x1)) -> 7(x1)
, 4(x1) -> 9(6(6(x1)))
, 2(4(x1)) -> 0(7(x1))
, 7(0(x1)) -> 9(3(x1))
, 6(2(x1)) -> 7(7(x1))
, 0(3(x1)) -> 5(3(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 1_0(3) -> 3
, 1_0(4) -> 3
, 1_0(9) -> 3
, 8_0(3) -> 4
, 8_0(4) -> 4
, 8_0(9) -> 4
, 3_0(3) -> 9
, 3_0(4) -> 9
, 3_0(9) -> 9
, 5^#_0(3) -> 16
, 5^#_0(4) -> 16
, 5^#_0(9) -> 16
, 6^#_0(3) -> 21
, 6^#_0(4) -> 21
, 6^#_0(9) -> 21}
21)
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))
, 4^#(7(x1)) -> c_7()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {4^#(7(x1)) -> c_7()}
Weak Rules:
{ 7^#(2(x1)) -> c_10(4^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{4^#(7(x1)) -> c_7()}
and weakly orienting the rules
{ 7^#(2(x1)) -> c_10(4^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(7(x1)) -> c_7()}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [1]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ 4^#(7(x1)) -> c_7()
, 7^#(2(x1)) -> c_10(4^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
Details:
The given problem does not contain any strict rules
22)
{ 2^#(8(x1)) -> c_6(7^#(x1))
, 7^#(2(x1)) -> c_10(4^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {7^#(2(x1)) -> c_10(4^#(x1))}
Weak Rules: {2^#(8(x1)) -> c_6(7^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{7^#(2(x1)) -> c_10(4^#(x1))}
and weakly orienting the rules
{2^#(8(x1)) -> c_6(7^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{7^#(2(x1)) -> c_10(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [1] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [1]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ 7^#(2(x1)) -> c_10(4^#(x1))
, 2^#(8(x1)) -> c_6(7^#(x1))}
Details:
The given problem does not contain any strict rules
23)
{ 2^#(8(x1)) -> c_2(4^#(x1))
, 4^#(7(x1)) -> c_7()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {4^#(7(x1)) -> c_7()}
Weak Rules: {2^#(8(x1)) -> c_2(4^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{4^#(7(x1)) -> c_7()}
and weakly orienting the rules
{2^#(8(x1)) -> c_2(4^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{4^#(7(x1)) -> c_7()}
Details:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [1]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ 4^#(7(x1)) -> c_7()
, 2^#(8(x1)) -> c_2(4^#(x1))}
Details:
The given problem does not contain any strict rules
24)
{2^#(8(1(x1))) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {2^#(8(1(x1))) -> c_1()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{2^#(8(1(x1))) -> c_1()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(1(x1))) -> c_1()}
Details:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [1] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {2^#(8(1(x1))) -> c_1()}
Details:
The given problem does not contain any strict rules
25)
{2^#(7(x1)) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {2^#(7(x1)) -> c_0()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{2^#(7(x1)) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(7(x1)) -> c_0()}
Details:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [1] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {2^#(7(x1)) -> c_0()}
Details:
The given problem does not contain any strict rules
26)
{2^#(8(x1)) -> c_2(4^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {2^#(8(x1)) -> c_2(4^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_2(4^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_2(4^#(x1))}
Details:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
4^#(x1) = [1] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {2^#(8(x1)) -> c_2(4^#(x1))}
Details:
The given problem does not contain any strict rules
27)
{2^#(8(x1)) -> c_6(7^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
7^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {2^#(8(x1)) -> c_6(7^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{2^#(8(x1)) -> c_6(7^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2^#(8(x1)) -> c_6(7^#(x1))}
Details:
Interpretation Functions:
2(x1) = [0] x1 + [0]
7(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
8(x1) = [1] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
9(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
3(x1) = [0] x1 + [0]
6(x1) = [0] x1 + [0]
2^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
4^#(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
6^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
7^#(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
9^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
c_17(x1) = [0] x1 + [0]
c_18() = [0]
c_19(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {2^#(8(x1)) -> c_6(7^#(x1))}
Details:
The given problem does not contain any strict rules