'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a__f(f(a())) -> a__f(g(f(a())))
, mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(X) -> f(X)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark^#(a()) -> c_2()
, mark^#(g(X)) -> c_3()
, a__f^#(X) -> c_4()}
The usable rules are:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)}
The estimated dependency graph contains the following edges:
{a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
==> {a__f^#(X) -> c_4()}
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
==> {a__f^#(X) -> c_4()}
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
==> {a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
We consider the following path(s):
1) { mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
, a__f^#(X) -> c_4()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(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:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(X) -> c_4()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f^#(X) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f^#(X) -> c_4()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
and weakly orienting the rules
{ mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f^#(X) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [13]
c_0(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(f(X)) -> a__f(mark(X))}
and weakly orienting the rules
{ mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f^#(X) -> c_4()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(f(X)) -> a__f(mark(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
a() = [1]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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:
{ a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
Weak Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f^#(X) -> c_4()}
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:
{ a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
Weak Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f^#(X) -> c_4()}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ a__f_0(10) -> 10
, a__f_1(12) -> 10
, a__f_1(16) -> 16
, a__f_2(18) -> 16
, f_0(2) -> 2
, f_0(3) -> 2
, f_0(4) -> 2
, f_1(10) -> 10
, f_1(14) -> 13
, f_2(12) -> 10
, f_2(16) -> 16
, f_2(20) -> 19
, f_3(18) -> 16
, a_0() -> 3
, a_0() -> 10
, a_1() -> 14
, a_1() -> 16
, a_2() -> 20
, g_0(2) -> 4
, g_0(2) -> 10
, g_0(3) -> 4
, g_0(3) -> 10
, g_0(4) -> 4
, g_0(4) -> 10
, g_1(2) -> 16
, g_1(3) -> 16
, g_1(4) -> 16
, g_1(13) -> 12
, g_2(19) -> 18
, mark_0(2) -> 10
, mark_0(3) -> 10
, mark_0(4) -> 10
, mark_1(2) -> 16
, mark_1(3) -> 16
, mark_1(4) -> 16
, a__f^#_0(2) -> 6
, a__f^#_0(3) -> 6
, a__f^#_0(4) -> 6
, a__f^#_0(10) -> 9
, a__f^#_1(12) -> 11
, a__f^#_1(16) -> 15
, a__f^#_2(18) -> 17
, c_0_1(11) -> 6
, c_0_1(11) -> 9
, c_0_2(17) -> 15
, mark^#_0(2) -> 8
, mark^#_0(3) -> 8
, mark^#_0(4) -> 8
, c_1_0(9) -> 8
, c_1_1(15) -> 8
, c_4_0() -> 6
, c_4_0() -> 9
, c_4_1() -> 11
, c_4_1() -> 15
, c_4_2() -> 17}
2) { mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(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:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
and weakly orienting the rules
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [3]
mark^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(f(X)) -> a__f(mark(X))}
and weakly orienting the rules
{ mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(f(X)) -> a__f(mark(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
a() = [0]
g(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [7]
a__f^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [8]
mark^#(x1) = [1] x1 + [13]
c_1(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4() = [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:
{ a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
Weak Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(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:
{ a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
Weak Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(X)}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ a__f_0(4) -> 4
, a__f_1(6) -> 4
, a__f_1(10) -> 10
, a__f_2(12) -> 10
, f_0(2) -> 2
, f_1(4) -> 4
, f_1(8) -> 7
, f_2(6) -> 4
, f_2(10) -> 10
, f_2(14) -> 13
, f_3(12) -> 10
, a_0() -> 2
, a_0() -> 4
, a_1() -> 8
, a_1() -> 10
, a_2() -> 14
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 10
, g_1(7) -> 6
, g_2(13) -> 12
, mark_0(2) -> 4
, mark_1(2) -> 10
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(6) -> 5
, a__f^#_1(10) -> 9
, a__f^#_2(12) -> 11
, c_0_1(5) -> 1
, c_0_1(5) -> 3
, c_0_2(11) -> 9
, mark^#_0(2) -> 1
, c_1_0(3) -> 1
, c_1_1(9) -> 1}
3) { mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(X) -> c_4()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(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:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(X) -> c_4()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f^#(X) -> c_4()}
and weakly orienting the rules
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(X) -> c_4()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [3]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
and weakly orienting the rules
{ a__f^#(X) -> c_4()
, mark(a()) -> a()
, mark(g(X)) -> g(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(X) -> f(X)}
and weakly orienting the rules
{ mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(X) -> c_4()
, mark(a()) -> a()
, mark(g(X)) -> g(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [8]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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:
{ mark(f(X)) -> a__f(mark(X))
, a__f(f(a())) -> a__f(g(f(a())))}
Weak Rules:
{ a__f(X) -> f(X)
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(X) -> c_4()
, mark(a()) -> a()
, mark(g(X)) -> g(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:
{ mark(f(X)) -> a__f(mark(X))
, a__f(f(a())) -> a__f(g(f(a())))}
Weak Rules:
{ a__f(X) -> f(X)
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, a__f^#(X) -> c_4()
, mark(a()) -> a()
, mark(g(X)) -> g(X)}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__f_1(11) -> 10
, a__f_1(11) -> 11
, a__f_2(13) -> 10
, a__f_2(13) -> 11
, f_0(2) -> 2
, f_0(3) -> 2
, f_0(4) -> 2
, f_1(11) -> 10
, f_1(11) -> 11
, f_2(13) -> 10
, f_2(13) -> 11
, f_2(15) -> 14
, a_0() -> 3
, a_0() -> 10
, a_1() -> 11
, a_2() -> 15
, g_0(2) -> 4
, g_0(2) -> 10
, g_0(3) -> 4
, g_0(3) -> 10
, g_0(4) -> 4
, g_0(4) -> 10
, g_1(2) -> 11
, g_1(3) -> 11
, g_1(4) -> 11
, g_2(14) -> 13
, mark_0(2) -> 10
, mark_0(3) -> 10
, mark_0(4) -> 10
, mark_1(2) -> 11
, mark_1(3) -> 11
, mark_1(4) -> 11
, a__f^#_0(2) -> 6
, a__f^#_0(3) -> 6
, a__f^#_0(4) -> 6
, a__f^#_0(10) -> 9
, a__f^#_1(11) -> 12
, mark^#_0(2) -> 8
, mark^#_0(3) -> 8
, mark^#_0(4) -> 8
, c_1_0(9) -> 8
, c_1_1(12) -> 8
, c_4_0() -> 6
, c_4_0() -> 9
, c_4_1() -> 12}
4) {mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(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:
{ mark(f(X)) -> a__f(mark(X))
, mark(a()) -> a()
, mark(g(X)) -> g(X)
, a__f(f(a())) -> a__f(g(f(a())))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
and weakly orienting the rules
{ mark(a()) -> a()
, mark(g(X)) -> g(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_1(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(X) -> f(X)}
and weakly orienting the rules
{ mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [2]
g(x1) = [1] x1 + [2]
mark(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [13]
c_1(x1) = [1] x1 + [8]
c_2() = [0]
c_3() = [0]
c_4() = [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:
{ mark(f(X)) -> a__f(mark(X))
, a__f(f(a())) -> a__f(g(f(a())))}
Weak Rules:
{ a__f(X) -> f(X)
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(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:
{ mark(f(X)) -> a__f(mark(X))
, a__f(f(a())) -> a__f(g(f(a())))}
Weak Rules:
{ a__f(X) -> f(X)
, mark^#(f(X)) -> c_1(a__f^#(mark(X)))
, mark(a()) -> a()
, mark(g(X)) -> g(X)}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__f_1(11) -> 10
, a__f_1(11) -> 11
, a__f_2(13) -> 10
, a__f_2(13) -> 11
, f_0(2) -> 2
, f_0(3) -> 2
, f_0(4) -> 2
, f_1(11) -> 10
, f_1(11) -> 11
, f_2(13) -> 10
, f_2(13) -> 11
, f_2(15) -> 14
, a_0() -> 3
, a_0() -> 10
, a_1() -> 11
, a_2() -> 15
, g_0(2) -> 4
, g_0(2) -> 10
, g_0(3) -> 4
, g_0(3) -> 10
, g_0(4) -> 4
, g_0(4) -> 10
, g_1(2) -> 11
, g_1(3) -> 11
, g_1(4) -> 11
, g_2(14) -> 13
, mark_0(2) -> 10
, mark_0(3) -> 10
, mark_0(4) -> 10
, mark_1(2) -> 11
, mark_1(3) -> 11
, mark_1(4) -> 11
, a__f^#_0(2) -> 6
, a__f^#_0(3) -> 6
, a__f^#_0(4) -> 6
, a__f^#_0(10) -> 9
, a__f^#_1(11) -> 12
, mark^#_0(2) -> 8
, mark^#_0(3) -> 8
, mark^#_0(4) -> 8
, c_1_0(9) -> 8
, c_1_1(12) -> 8}
5) {mark^#(g(X)) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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: {mark^#(g(X)) -> c_3()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(g(X)) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(g(X)) -> c_3()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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: {mark^#(g(X)) -> c_3()}
Details:
The given problem does not contain any strict rules
6) {mark^#(a()) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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: {mark^#(a()) -> c_2()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(a()) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(a()) -> c_2()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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: {mark^#(a()) -> c_2()}
Details:
The given problem does not contain any strict rules