'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, d(c(x1)) -> b(b(b(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, d^#(b(c(x1))) -> c_6()
, d^#(a(x1)) -> c_7(b^#(x1))}
The usable rules are:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)}
The estimated dependency graph contains the following edges:
{b^#(c(a(x1))) -> c_0(b^#(x1))}
==> {b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
{b^#(c(a(x1))) -> c_0(b^#(x1))}
==> {b^#(c(a(x1))) -> c_0(b^#(x1))}
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
==> {d^#(a(x1)) -> c_7(b^#(x1))}
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
==> {d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
==> {d^#(b(c(x1))) -> c_6()}
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
==> {d^#(a(x1)) -> c_7(b^#(x1))}
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
==> {d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
==> {d^#(b(c(x1))) -> c_6()}
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
==> {b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
==> {b^#(c(a(x1))) -> c_0(b^#(x1))}
{c^#(b(x1)) -> c_5(d^#(a(x1)))}
==> {d^#(a(x1)) -> c_7(b^#(x1))}
{d^#(a(x1)) -> c_7(b^#(x1))}
==> {b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
{d^#(a(x1)) -> c_7(b^#(x1))}
==> {b^#(c(a(x1))) -> c_0(b^#(x1))}
We consider the following path(s):
1) { c^#(b(x1)) -> c_5(d^#(a(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_5(d^#(a(x1)))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_5(d^#(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{ c^#(b(x1)) -> c_5(d^#(a(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(a(x1))) -> c_0(b^#(x1))}
and weakly orienting the rules
{ d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [6]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [5]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [6]
c_0(x1) = [1] x1 + [5]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [6]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{ d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
2) { c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [3]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [5]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [6]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [2]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [15]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [2]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{ d(a(x1)) -> b(x1)
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [5]
c(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [11]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ d^#(a(x1)) -> c_7(b^#(x1))
, d(a(x1)) -> b(x1)
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, d(a(x1)) -> b(x1)
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, d(a(x1)) -> b(x1)
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
3) { c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [7]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [2]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ d^#(a(x1)) -> c_7(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [13]
c_0(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [2]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
4) { c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(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:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [4]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [6]
b^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [11]
c_2(x1) = [1] x1 + [8]
d^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [8]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
and weakly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [5]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [7]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [14]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
5) { c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [3]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [2]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [7]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{ d(a(x1)) -> b(x1)
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [3]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ d^#(a(x1)) -> c_7(b^#(x1))
, d(a(x1)) -> b(x1)
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [6]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [5]
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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, d(a(x1)) -> b(x1)
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, d(a(x1)) -> b(x1)
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
6) { c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(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:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [6]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [6]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [7]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [3]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
7) { c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(a(x1))) -> c_0(b^#(x1))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [15]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [2]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
8) { c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [7]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [3]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [12]
d^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
9) { c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(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:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [4]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [6]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [13]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [6]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
10)
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(a(x1))) -> c_0(b^#(x1))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [15]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [3]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [3]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [3]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
11)
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(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:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [4]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [7]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [10]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
12)
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(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:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [7]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [3]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(c(a(x1))) -> a(b(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [10]
c(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [10]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [2]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ b(c(a(x1))) -> a(b(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ b(c(a(x1))) -> a(b(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
13)
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(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:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [6]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [7]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [5]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [6]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
14)
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(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:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [10]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [10]
c_2(x1) = [1] x1 + [4]
d^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ d(a(x1)) -> b(x1)
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(c(a(x1))) -> a(b(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
15)
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
The usable rules for this path are the following:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(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:
{ b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [3]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(c(x1)) -> c_4(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [4]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(c(a(x1))) -> a(b(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))}
and weakly orienting the rules
{ d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(c(a(x1))) -> a(b(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [11]
c(x1) = [1] x1 + [10]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [10]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [15]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ b(c(a(x1))) -> a(b(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(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:
{ c(d(x1)) -> d(c(x1))
, d(c(x1)) -> b(b(b(x1)))}
Weak Rules:
{ b(c(a(x1))) -> a(b(x1))
, d(a(x1)) -> b(x1)
, b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(c(x1)) -> c_4(b^#(b(b(x1))))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, b(b(b(x1))) -> c(a(c(x1)))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
16)
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(a(x1)) -> c_7(b^#(x1))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [6]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [3]
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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
17)
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(b(c(x1))) -> c_6()}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, d^#(b(c(x1))) -> c_6()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [11]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [15]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ d(a(x1)) -> b(x1)
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
18)
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [2]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
19)
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(b(x1))) -> c_3(d^#(c(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
20)
{ c^#(b(x1)) -> c_5(d^#(a(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [3]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))}
and weakly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, b^#(b(b(x1))) -> c_1(c^#(a(c(x1))))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, d^#(a(x1)) -> c_7(b^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, b^#_0(3) -> 5
, c^#_0(3) -> 8
, d^#_0(3) -> 10
, c_7_0(5) -> 10}
21)
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(b(c(x1))) -> c_6()}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, d^#(b(c(x1))) -> c_6()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(a(x1)) -> b(x1)}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(a(x1)) -> b(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ d(a(x1)) -> b(x1)
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, d(a(x1)) -> b(x1)
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, b(c(a(x1))) -> a(b(x1))
, d^#(b(c(x1))) -> c_6()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
22)
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
The usable rules for this path are the following:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(c(x1)) -> b(b(b(x1)))
, d(b(c(x1))) -> a(a(x1))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))
, b(b(b(x1))) -> c(a(c(x1)))
, c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
and weakly orienting the rules
{ d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_2(d^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_2(d^#(c(x1)))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [5]
c_2(x1) = [1] x1 + [4]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(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:
{ c(d(x1)) -> d(c(x1))
, c(d(b(x1))) -> d(c(c(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(a(c(x1)))
, c(b(x1)) -> d(a(x1))
, d(b(c(x1))) -> a(a(x1))
, c^#(d(x1)) -> c_2(d^#(c(x1)))
, d(c(x1)) -> b(b(b(x1)))
, d(a(x1)) -> b(x1)
, b(c(a(x1))) -> a(b(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(3) -> 3
, c^#_0(3) -> 8
, d^#_0(3) -> 10}
23)
{ c^#(b(x1)) -> c_5(d^#(a(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))
, b^#(c(a(x1))) -> c_0(b^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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: {b^#(c(a(x1))) -> c_0(b^#(x1))}
Weak Rules:
{ d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(c(a(x1))) -> c_0(b^#(x1))}
and weakly orienting the rules
{ d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(a(x1))) -> c_0(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [6]
c(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [7]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
c_6() = [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ b^#(c(a(x1))) -> c_0(b^#(x1))
, d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))}
Details:
The given problem does not contain any strict rules
24)
{ c^#(b(x1)) -> c_5(d^#(a(x1)))
, d^#(a(x1)) -> c_7(b^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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: {d^#(a(x1)) -> c_7(b^#(x1))}
Weak Rules: {c^#(b(x1)) -> c_5(d^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x1)) -> c_7(b^#(x1))}
and weakly orienting the rules
{c^#(b(x1)) -> c_5(d^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x1)) -> c_7(b^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [0] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [1] 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:
{ d^#(a(x1)) -> c_7(b^#(x1))
, c^#(b(x1)) -> c_5(d^#(a(x1)))}
Details:
The given problem does not contain any strict rules
25)
{c^#(b(x1)) -> c_5(d^#(a(x1)))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(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: {c^#(b(x1)) -> c_5(d^#(a(x1)))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_5(d^#(a(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_5(d^#(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [0] x1 + [0]
a(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(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: {c^#(b(x1)) -> c_5(d^#(a(x1)))}
Details:
The given problem does not contain any strict rules