'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(a(a(x))) -> c_0()
, b^#(u(x)) -> c_1(b^#(d(x)))
, d^#(a(x)) -> c_2(a^#(d(x)))
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(u(x)) -> c_4(a^#(x))}
The usable rules are:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
The estimated dependency graph contains the following edges:
{b^#(u(x)) -> c_1(b^#(d(x)))}
==> {b^#(u(x)) -> c_1(b^#(d(x)))}
{d^#(a(x)) -> c_2(a^#(d(x)))}
==> {a^#(u(x)) -> c_4(a^#(x))}
{d^#(a(x)) -> c_2(a^#(d(x)))}
==> {a^#(a(a(x))) -> c_0()}
{d^#(b(x)) -> c_3(a^#(b(x)))}
==> {a^#(u(x)) -> c_4(a^#(x))}
{d^#(b(x)) -> c_3(a^#(b(x)))}
==> {a^#(a(a(x))) -> c_0()}
{a^#(u(x)) -> c_4(a^#(x))}
==> {a^#(u(x)) -> c_4(a^#(x))}
{a^#(u(x)) -> c_4(a^#(x))}
==> {a^#(a(a(x))) -> c_0()}
We consider the following path(s):
1) { d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(u(x)) -> c_4(a^#(x))
, a^#(a(a(x))) -> c_0()}
The usable rules for this path are the following:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, a^#(u(x)) -> c_4(a^#(x))
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(a(x))) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(a(x))) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [7]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(b(x)) -> c_3(a^#(b(x)))}
and weakly orienting the rules
{a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(b(x)) -> c_3(a^#(b(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{ d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [2]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(b(x)) -> u(a(b(x)))}
and weakly orienting the rules
{ a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(b(x)) -> u(a(b(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [4]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [11]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(u(x)) -> c_4(a^#(x))}
and weakly orienting the rules
{ d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(u(x)) -> c_4(a^#(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [15]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(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:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a^#(u(x)) -> c_4(a^#(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
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:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a^#(u(x)) -> c_4(a^#(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 2
, a^#_0(2) -> 1
, d^#_0(2) -> 1
, c_4_0(1) -> 1}
2) { d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(u(x)) -> c_4(a^#(x))
, a^#(a(a(x))) -> c_0()}
The usable rules for this path are the following:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, a^#(u(x)) -> c_4(a^#(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x)) -> c_2(a^#(d(x)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x)) -> c_2(a^#(d(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [9]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(a(a(x))) -> c_0()}
and weakly orienting the rules
{d^#(a(x)) -> c_2(a^#(d(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(a(x))) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [8]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(u(x)) -> c_4(a^#(x))}
and weakly orienting the rules
{ a^#(a(a(x))) -> c_0()
, d^#(a(x)) -> c_2(a^#(d(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(u(x)) -> c_4(a^#(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [8]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(u(x)) -> b(d(x))}
and weakly orienting the rules
{ a^#(u(x)) -> c_4(a^#(x))
, a^#(a(a(x))) -> c_0()
, d^#(a(x)) -> c_2(a^#(d(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(u(x)) -> b(d(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [9]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [3]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{ b(u(x)) -> b(d(x))
, a^#(u(x)) -> c_4(a^#(x))
, a^#(a(a(x))) -> c_0()
, d^#(a(x)) -> c_2(a^#(d(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
c() = [0]
b(x1) = [1] x1 + [4]
u(x1) = [1] x1 + [12]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, a^#(u(x)) -> c_4(a^#(x))
, a^#(a(a(x))) -> c_0()
, d^#(a(x)) -> c_2(a^#(d(x)))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, a^#(u(x)) -> c_4(a^#(x))
, a^#(a(a(x))) -> c_0()
, d^#(a(x)) -> c_2(a^#(d(x)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 4
, u_0(4) -> 4
, a^#_0(2) -> 6
, a^#_0(4) -> 6
, d^#_0(2) -> 10
, d^#_0(4) -> 10
, c_4_0(6) -> 6}
3) { d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(u(x)) -> c_4(a^#(x))}
The usable rules for this path are the following:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(u(x)) -> c_4(a^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(u(x)) -> c_4(a^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(u(x)) -> c_4(a^#(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x)) -> c_2(a^#(d(x)))}
and weakly orienting the rules
{a^#(u(x)) -> c_4(a^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x)) -> c_2(a^#(d(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(u(x)) -> b(d(x))}
and weakly orienting the rules
{ d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(u(x)) -> c_4(a^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(u(x)) -> b(d(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [9]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{ b(u(x)) -> b(d(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(u(x)) -> c_4(a^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(u(x)) -> c_4(a^#(x))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(u(x)) -> c_4(a^#(x))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 4
, u_0(4) -> 4
, a^#_0(2) -> 6
, a^#_0(4) -> 6
, d^#_0(2) -> 10
, d^#_0(4) -> 10
, c_4_0(6) -> 6}
4) { d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
The usable rules for this path are the following:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#(a(a(x))) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(a(x))) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(b(x)) -> c_3(a^#(b(x)))}
and weakly orienting the rules
{a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(b(x)) -> c_3(a^#(b(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{ d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(u(x)) -> b(d(x))}
and weakly orienting the rules
{ a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(u(x)) -> b(d(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [12]
u(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(a(a(x))) -> c_0()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 4
, u_0(4) -> 4
, a^#_0(2) -> 6
, a^#_0(4) -> 6
, d^#_0(2) -> 10
, d^#_0(4) -> 10}
5) { d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
The usable rules for this path are the following:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [2]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(u(x)) -> b(d(x))}
and weakly orienting the rules
{ d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(u(x)) -> b(d(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [9]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [4]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{ b(u(x)) -> b(d(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [4]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [5]
c_2(x1) = [1] x1 + [5]
c_3(x1) = [0] x1 + [0]
c_4(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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, d^#(a(x)) -> c_2(a^#(d(x)))
, a^#(a(a(x))) -> c_0()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 4
, u_0(4) -> 4
, a^#_0(2) -> 6
, a^#_0(4) -> 6
, d^#_0(2) -> 10
, d^#_0(4) -> 10}
6) { d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(u(x)) -> c_4(a^#(x))}
The usable rules for this path are the following:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, d^#(b(x)) -> c_3(a^#(b(x)))
, a^#(u(x)) -> c_4(a^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(b(x)) -> c_3(a^#(b(x)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(b(x)) -> c_3(a^#(b(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [14]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{d^#(b(x)) -> c_3(a^#(b(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(u(x)) -> b(d(x))
, a^#(u(x)) -> c_4(a^#(x))}
and weakly orienting the rules
{ a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(u(x)) -> b(d(x))
, a^#(u(x)) -> c_4(a^#(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [14]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ b(u(x)) -> b(d(x))
, a^#(u(x)) -> c_4(a^#(x))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ b(u(x)) -> b(d(x))
, a^#(u(x)) -> c_4(a^#(x))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 4
, u_0(4) -> 4
, a^#_0(2) -> 6
, a^#_0(4) -> 6
, d^#_0(2) -> 10
, d^#_0(4) -> 10
, c_4_0(6) -> 6}
7) {d^#(b(x)) -> c_3(a^#(b(x)))}
The usable rules for this path are the following:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, d^#(b(x)) -> c_3(a^#(b(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(b(x)) -> c_3(a^#(b(x)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(b(x)) -> c_3(a^#(b(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{d^#(b(x)) -> c_3(a^#(b(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [12]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(b(x)) -> u(a(b(x)))}
and weakly orienting the rules
{ a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(b(x)) -> u(a(b(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [1]
u(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(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:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))}
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:
{ b(u(x)) -> b(d(x))
, d(a(x)) -> a(d(x))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ d(b(x)) -> u(a(b(x)))
, a(a(a(x))) -> c()
, d^#(b(x)) -> c_3(a^#(b(x)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 4
, u_0(4) -> 4
, a^#_0(2) -> 6
, a^#_0(4) -> 6
, d^#_0(2) -> 10
, d^#_0(4) -> 10}
8) {d^#(a(x)) -> c_2(a^#(d(x)))}
The usable rules for this path are the following:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, d^#(a(x)) -> c_2(a^#(d(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b(u(x)) -> b(d(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(u(x)) -> b(d(x))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [5]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(a(x)) -> c_2(a^#(d(x)))}
and weakly orienting the rules
{b(u(x)) -> b(d(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(a(x)) -> c_2(a^#(d(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{ d^#(a(x)) -> c_2(a^#(d(x)))
, b(u(x)) -> b(d(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0() = [0]
b^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [5]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, d^#(a(x)) -> c_2(a^#(d(x)))
, b(u(x)) -> b(d(x))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, d^#(a(x)) -> c_2(a^#(d(x)))
, b(u(x)) -> b(d(x))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 4
, u_0(4) -> 4
, a^#_0(2) -> 6
, a^#_0(4) -> 6
, d^#_0(2) -> 10
, d^#_0(4) -> 10}
9) {b^#(u(x)) -> c_1(b^#(d(x)))}
The usable rules for this path are the following:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, b(u(x)) -> b(d(x))
, a(a(a(x))) -> c()
, a(u(x)) -> u(a(x))
, b^#(u(x)) -> c_1(b^#(d(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(u(x)) -> b(d(x))
, b^#(u(x)) -> c_1(b^#(d(x)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(u(x)) -> b(d(x))
, b^#(u(x)) -> c_1(b^#(d(x)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [9]
d(x1) = [1] x1 + [1]
a^#(x1) = [0] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(a(a(x))) -> c()}
and weakly orienting the rules
{ b(u(x)) -> b(d(x))
, b^#(u(x)) -> c_1(b^#(d(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(a(a(x))) -> c()}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
c() = [0]
b(x1) = [1] x1 + [0]
u(x1) = [1] x1 + [9]
d(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0() = [0]
b^#(x1) = [1] x1 + [13]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, b^#(u(x)) -> c_1(b^#(d(x)))}
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:
{ d(a(x)) -> a(d(x))
, d(b(x)) -> u(a(b(x)))
, a(u(x)) -> u(a(x))}
Weak Rules:
{ a(a(a(x))) -> c()
, b(u(x)) -> b(d(x))
, b^#(u(x)) -> c_1(b^#(d(x)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0() -> 2
, u_0(2) -> 2
, d_0(2) -> 4
, b^#_0(2) -> 1
, b^#_0(4) -> 3
, c_1_0(3) -> 1}