'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ g(h(x1)) -> g(f(s(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, b(a(x1)) -> a(b(x1))
, a(a(a(x1))) -> b(a(a(b(x1))))
, b(b(b(b(x1)))) -> a(x1)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ g^#(h(x1)) -> c_0(g^#(f(s(x1))))
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, b^#(a(x1)) -> c_5(a^#(b(x1)))
, a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))
, b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
The usable rules are:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, b(a(x1)) -> a(b(x1))
, a(a(a(x1))) -> b(a(a(b(x1))))
, b(b(b(b(x1)))) -> a(x1)}
The estimated dependency graph contains the following edges:
{g^#(h(x1)) -> c_0(g^#(f(s(x1))))}
==> {g^#(h(x1)) -> c_0(g^#(f(s(x1))))}
{f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
==> {h^#(x1) -> c_3()}
{f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
==> {h^#(x1) -> c_3()}
{f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
==> {f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
{f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
==> {f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
{f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
==> {f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
{b^#(a(x1)) -> c_5(a^#(b(x1)))}
==> {a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))}
{a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))}
==> {b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
{a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))}
==> {b^#(a(x1)) -> c_5(a^#(b(x1)))}
{b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
==> {a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))}
We consider the following path(s):
1) { f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
The usable rules for this path are the following:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(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:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, h^#(x1) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{h^#(x1) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(x1) -> c_3()}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [7]
h^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
and weakly orienting the rules
{h^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [7]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [11]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(x1) -> x1}
and weakly orienting the rules
{ f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(x1) -> x1}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [4]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [2]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
and weakly orienting the rules
{ h(x1) -> x1
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [3]
h^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, h(x1) -> x1
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, h(x1) -> x1
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ h_0(4) -> 15
, h_0(13) -> 13
, f_0(14) -> 13
, s_0(4) -> 4
, s_0(4) -> 15
, s_0(15) -> 14
, f^#_0(4) -> 9
, c_1_0(12) -> 9
, h^#_0(4) -> 11
, h^#_0(13) -> 12
, c_3_0() -> 11
, c_3_0() -> 12}
2) { f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
The usable rules for this path are the following:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(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:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, h^#(x1) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{h^#(x1) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(x1) -> c_3()}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [7]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
and weakly orienting the rules
{h^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(x1) -> x1}
and weakly orienting the rules
{ f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(x1) -> x1}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [2]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [4]
c_3() = [0]
c_4(x1) = [1] x1 + [5]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
and weakly orienting the rules
{ h(x1) -> x1
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [1]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, h(x1) -> x1
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, h(x1) -> x1
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))
, h^#(x1) -> c_3()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ s_0(4) -> 4
, f^#_0(4) -> 9
, h^#_0(4) -> 11
, c_3_0() -> 11}
3) { b^#(a(x1)) -> c_5(a^#(b(x1)))
, a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))
, b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
The usable rules for this path are the following:
{ b(a(x1)) -> a(b(x1))
, a(a(a(x1))) -> b(a(a(b(x1))))
, b(b(b(b(x1)))) -> a(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(a(x1)) -> a(b(x1))
, a(a(a(x1))) -> b(a(a(b(x1))))
, b(b(b(b(x1)))) -> a(x1)
, b^#(a(x1)) -> c_5(a^#(b(x1)))
, a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))
, b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(b(x1)))) -> a(x1)
, b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(b(x1)))) -> a(x1)
, b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))}
and weakly orienting the rules
{ b(b(b(b(x1)))) -> a(x1)
, b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [4]
a^#(x1) = [1] x1 + [4]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(a(a(x1))) -> b(a(a(b(x1))))
, b^#(a(x1)) -> c_5(a^#(b(x1)))}
and weakly orienting the rules
{ a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))
, b(b(b(b(x1)))) -> a(x1)
, b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(a(x1))) -> b(a(a(b(x1))))
, b^#(a(x1)) -> c_5(a^#(b(x1)))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
b(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [12]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [13]
c_6(x1) = [1] x1 + [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: {b(a(x1)) -> a(b(x1))}
Weak Rules:
{ a(a(a(x1))) -> b(a(a(b(x1))))
, b^#(a(x1)) -> c_5(a^#(b(x1)))
, a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))
, b(b(b(b(x1)))) -> a(x1)
, b^#(b(b(b(x1)))) -> c_7(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: {b(a(x1)) -> a(b(x1))}
Weak Rules:
{ a(a(a(x1))) -> b(a(a(b(x1))))
, b^#(a(x1)) -> c_5(a^#(b(x1)))
, a^#(a(a(x1))) -> c_6(b^#(a(a(b(x1)))))
, b(b(b(b(x1)))) -> a(x1)
, b^#(b(b(b(x1)))) -> c_7(a^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b^#_0(20) -> 15
, a^#_0(20) -> 17}
4) { f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
The usable rules for this path are the following:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(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:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(x1) -> x1}
and weakly orienting the rules
{f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(x1) -> x1}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [8]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [15]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(s(s(s(x1)))) -> h(f(s(h(x1))))}
and weakly orienting the rules
{ h(x1) -> x1
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(s(s(s(x1)))) -> h(f(s(h(x1))))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [1]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [4]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [2]
c_3() = [0]
c_4(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
and weakly orienting the rules
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, h(x1) -> x1
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [4]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, h(x1) -> x1
, f^#(h(x1)) -> c_2(h^#(f(s(h(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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, h(x1) -> x1
, f^#(h(x1)) -> c_2(h^#(f(s(h(x1)))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ s_0(4) -> 4
, f^#_0(4) -> 9
, h^#_0(4) -> 11}
5) { f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
The usable rules for this path are the following:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(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:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(x1) -> x1}
and weakly orienting the rules
{f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(x1) -> x1}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [6]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
and weakly orienting the rules
{ h(x1) -> x1
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [1]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, h(x1) -> x1
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, h(x1) -> x1
, f^#(s(s(s(x1)))) -> c_1(h^#(f(s(h(x1)))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ h_0(4) -> 15
, h_0(13) -> 13
, f_0(14) -> 13
, s_0(4) -> 4
, s_0(4) -> 15
, s_0(15) -> 14
, f^#_0(4) -> 9
, c_1_0(12) -> 9
, h^#_0(4) -> 11
, h^#_0(13) -> 12}
6) {f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
The usable rules for this path are the following:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, h(x1) -> 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:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, h(x1) -> x1
, f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{h(x1) -> x1}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(x1) -> x1}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [7]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(s(s(s(x1)))) -> h(f(s(h(x1))))}
and weakly orienting the rules
{h(x1) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(s(s(s(x1)))) -> h(f(s(h(x1))))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [1]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [15]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
and weakly orienting the rules
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, h(x1) -> x1}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [8]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [6]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, h(x1) -> 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:
{ f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ f^#(f(s(s(x1)))) -> c_4(f^#(f(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, h(x1) -> x1}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ s_0(4) -> 4
, f^#_0(4) -> 9}
7) {g^#(h(x1)) -> c_0(g^#(f(s(x1))))}
The usable rules for this path are the following:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, h(x1) -> 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:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, h(x1) -> x1
, g^#(h(x1)) -> c_0(g^#(f(s(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ h(x1) -> x1
, g^#(h(x1)) -> c_0(g^#(f(s(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ h(x1) -> x1
, g^#(h(x1)) -> c_0(g^#(f(s(x1))))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [8]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
b(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [3]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ h(x1) -> x1
, g^#(h(x1)) -> c_0(g^#(f(s(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:
{ f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))}
Weak Rules:
{ h(x1) -> x1
, g^#(h(x1)) -> c_0(g^#(f(s(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ s_0(4) -> 4
, g^#_0(4) -> 7}