'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, a__c^#(X) -> c_1()
, a__h^#(X) -> c_2(a__c^#(d(X)))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(c(X)) -> c_4(a__c^#(X))
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, mark^#(g(X)) -> c_6()
, mark^#(d(X)) -> c_7()
, a__f^#(X) -> c_8()
, a__c^#(X) -> c_9()
, a__h^#(X) -> c_10()}
The usable rules are:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)}
The estimated dependency graph contains the following edges:
{a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
==> {a__c^#(X) -> c_9()}
{a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
==> {a__c^#(X) -> c_1()}
{a__h^#(X) -> c_2(a__c^#(d(X)))}
==> {a__c^#(X) -> c_9()}
{a__h^#(X) -> c_2(a__c^#(d(X)))}
==> {a__c^#(X) -> c_1()}
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
==> {a__f^#(X) -> c_8()}
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
==> {a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
{mark^#(c(X)) -> c_4(a__c^#(X))}
==> {a__c^#(X) -> c_9()}
{mark^#(c(X)) -> c_4(a__c^#(X))}
==> {a__c^#(X) -> c_1()}
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
==> {a__h^#(X) -> c_10()}
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
==> {a__h^#(X) -> c_2(a__c^#(d(X)))}
We consider the following path(s):
1) { mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, a__c^#(X) -> c_9()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__c^#(X) -> c_9()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__c^#(X) -> c_9()}
and weakly orienting the rules
{ a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#(X) -> c_9()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [4]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ a__c^#(X) -> c_9()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_9()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_9()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_9()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [9]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_9()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_9()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, f_0(2) -> 2
, f_1(2) -> 11
, f_1(5) -> 4
, f_1(5) -> 5
, f_1(10) -> 9
, f_1(14) -> 13
, a__c_0(2) -> 4
, a__c_1(2) -> 5
, a__c_1(6) -> 4
, a__c_1(6) -> 5
, a__c_1(13) -> 4
, a__c_1(13) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 5
, g_1(4) -> 14
, g_1(11) -> 10
, d_0(2) -> 2
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(5) -> 6
, d_1(6) -> 4
, d_1(6) -> 5
, d_1(13) -> 4
, d_1(13) -> 5
, a__h_1(5) -> 4
, a__h_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 5
, c_1(6) -> 4
, c_1(6) -> 5
, c_1(13) -> 4
, c_1(13) -> 5
, h_0(2) -> 2
, h_1(5) -> 4
, h_1(5) -> 5
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 7
, c_0_0(1) -> 1
, c_0_1(8) -> 1
, c_0_1(12) -> 3
, c_0_1(12) -> 7
, a__c^#_0(2) -> 1
, a__c^#_1(9) -> 8
, a__c^#_1(13) -> 12
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(7) -> 1
, c_9_0() -> 1
, c_9_1() -> 8
, c_9_1() -> 12}
2) { mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [3]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [7]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ mark(h(X)) -> a__h(mark(X))
, a__h(X) -> a__c(d(X))}
and weakly orienting the rules
{ a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(h(X)) -> a__h(mark(X))
, a__h(X) -> a__c(d(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [3]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [12]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ mark(h(X)) -> a__h(mark(X))
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [8]
a__f^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [3]
a__c^#(x1) = [1] x1 + [1]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(f(X)) -> a__c(f(g(f(X))))}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(f(X)) -> a__c(f(g(f(X))))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [8]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [8]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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__h(X) -> h(X)}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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__h(X) -> h(X)}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, f_0(2) -> 2
, f_1(2) -> 11
, f_1(5) -> 4
, f_1(5) -> 5
, f_1(10) -> 9
, f_1(14) -> 13
, f_2(5) -> 19
, f_2(18) -> 17
, a__c_0(2) -> 4
, a__c_1(2) -> 5
, a__c_1(6) -> 4
, a__c_2(15) -> 5
, a__c_2(17) -> 4
, a__c_2(17) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 5
, g_1(4) -> 14
, g_1(11) -> 10
, g_2(19) -> 18
, d_0(2) -> 2
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(4) -> 6
, d_1(5) -> 2
, d_1(5) -> 4
, d_1(5) -> 5
, d_1(6) -> 4
, d_2(5) -> 15
, d_2(15) -> 5
, d_2(17) -> 4
, d_2(17) -> 5
, a__h_0(4) -> 4
, a__h_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 5
, c_1(6) -> 4
, c_2(15) -> 5
, c_2(17) -> 4
, c_2(17) -> 5
, h_0(2) -> 2
, h_1(4) -> 4
, h_2(5) -> 5
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 7
, c_0_0(1) -> 1
, c_0_1(8) -> 1
, c_0_1(12) -> 3
, c_0_2(16) -> 7
, a__c^#_0(2) -> 1
, a__c^#_1(9) -> 8
, a__c^#_1(13) -> 12
, a__c^#_2(17) -> 16
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(7) -> 1}
3) { mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, a__c^#(X) -> c_1()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__c^#(X) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__c^#(X) -> c_1()}
and weakly orienting the rules
{ a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#(X) -> c_1()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [4]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ a__c^#(X) -> c_1()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [1]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_1()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_1()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_1()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [9]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_1()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(c(X)) -> a__c(X)
, a__c^#(X) -> c_1()
, a__f^#(f(X)) -> c_0(a__c^#(f(g(f(X)))))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, f_0(2) -> 2
, f_1(2) -> 11
, f_1(5) -> 4
, f_1(5) -> 5
, f_1(10) -> 9
, f_1(14) -> 13
, a__c_0(2) -> 4
, a__c_1(2) -> 5
, a__c_1(6) -> 4
, a__c_1(6) -> 5
, a__c_1(13) -> 4
, a__c_1(13) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 5
, g_1(4) -> 14
, g_1(11) -> 10
, d_0(2) -> 2
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(5) -> 6
, d_1(6) -> 4
, d_1(6) -> 5
, d_1(13) -> 4
, d_1(13) -> 5
, a__h_1(5) -> 4
, a__h_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 5
, c_1(6) -> 4
, c_1(6) -> 5
, c_1(13) -> 4
, c_1(13) -> 5
, h_0(2) -> 2
, h_1(5) -> 4
, h_1(5) -> 5
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 7
, c_0_0(1) -> 1
, c_0_1(8) -> 1
, c_0_1(12) -> 3
, c_0_1(12) -> 7
, a__c^#_0(2) -> 1
, a__c^#_1(9) -> 8
, a__c^#_1(13) -> 12
, c_1_0() -> 1
, c_1_1() -> 8
, c_1_1() -> 12
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(7) -> 1}
4) { mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_9()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__c^#(X) -> c_9()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__c^#(X) -> c_9()}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#(X) -> c_9()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [2]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__h^#(X) -> c_2(a__c^#(d(X)))}
and weakly orienting the rules
{ a__c^#(X) -> c_9()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(X) -> c_2(a__c^#(d(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [7]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(X) -> f(X)
, a__h(X) -> h(X)}
and weakly orienting the rules
{ a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_9()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(X) -> f(X)
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [3]
f(x1) = [1] x1 + [2]
a__c(x1) = [1] x1 + [5]
g(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [1]
mark(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [1]
c_1() = [0]
a__h^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [3]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
and weakly orienting the rules
{ a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_9()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_9()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__h(X) -> a__c(d(X))}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_9()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h(X) -> a__c(d(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(f(X)) -> a__c(f(g(f(X))))}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_9()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(f(X)) -> a__c(f(g(f(X))))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [4]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [1]
c_1() = [0]
a__h^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__h(X) -> a__c(d(X))
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_9()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__h(X) -> a__c(d(X))
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_9()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(7) -> 4
, a__f_1(7) -> 7
, f_0(2) -> 2
, f_1(7) -> 4
, f_1(7) -> 7
, f_1(15) -> 8
, a__c_0(2) -> 4
, a__c_1(2) -> 7
, a__c_1(8) -> 4
, a__c_1(8) -> 7
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 7
, g_1(7) -> 15
, d_0(2) -> 2
, d_0(2) -> 4
, d_0(4) -> 6
, d_1(2) -> 7
, d_1(2) -> 11
, d_1(4) -> 13
, d_1(7) -> 8
, d_1(8) -> 4
, d_1(8) -> 7
, a__h_1(7) -> 4
, a__h_1(7) -> 7
, mark_0(2) -> 4
, mark_1(2) -> 7
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 7
, c_1(8) -> 4
, c_1(8) -> 7
, h_0(2) -> 2
, h_1(7) -> 4
, h_1(7) -> 7
, a__c^#_0(2) -> 1
, a__c^#_0(6) -> 5
, a__c^#_1(8) -> 14
, a__c^#_1(11) -> 10
, a__c^#_1(13) -> 12
, a__h^#_0(2) -> 1
, a__h^#_0(4) -> 3
, a__h^#_1(7) -> 9
, c_2_0(1) -> 1
, c_2_0(5) -> 3
, c_2_1(10) -> 1
, c_2_1(12) -> 3
, c_2_1(14) -> 9
, mark^#_0(2) -> 1
, c_5_0(3) -> 1
, c_5_1(9) -> 1
, c_9_0() -> 1
, c_9_0() -> 5
, c_9_1() -> 10
, c_9_1() -> 12
, c_9_1() -> 14}
5) { mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_1()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__c^#(X) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__c^#(X) -> c_1()}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#(X) -> c_1()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [2]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__h^#(X) -> c_2(a__c^#(d(X)))}
and weakly orienting the rules
{ a__c^#(X) -> c_1()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(X) -> c_2(a__c^#(d(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [7]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(X) -> f(X)
, a__h(X) -> h(X)}
and weakly orienting the rules
{ a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_1()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(X) -> f(X)
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [3]
f(x1) = [1] x1 + [2]
a__c(x1) = [1] x1 + [5]
g(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [1]
mark(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [1]
c_1() = [0]
a__h^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [3]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
and weakly orienting the rules
{ a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_1()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_1()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__h(X) -> a__c(d(X))}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_1()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h(X) -> a__c(d(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(f(X)) -> a__c(f(g(f(X))))}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_1()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(f(X)) -> a__c(f(g(f(X))))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [4]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [1]
c_1() = [0]
a__h^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__h(X) -> a__c(d(X))
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_1()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__h(X) -> a__c(d(X))
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__f(X) -> f(X)
, a__h(X) -> h(X)
, a__h^#(X) -> c_2(a__c^#(d(X)))
, a__c^#(X) -> c_1()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(7) -> 4
, a__f_1(7) -> 7
, f_0(2) -> 2
, f_1(7) -> 4
, f_1(7) -> 7
, f_1(15) -> 8
, a__c_0(2) -> 4
, a__c_1(2) -> 7
, a__c_1(8) -> 4
, a__c_1(8) -> 7
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 7
, g_1(7) -> 15
, d_0(2) -> 2
, d_0(2) -> 4
, d_0(4) -> 6
, d_1(2) -> 7
, d_1(2) -> 11
, d_1(4) -> 13
, d_1(7) -> 8
, d_1(8) -> 4
, d_1(8) -> 7
, a__h_1(7) -> 4
, a__h_1(7) -> 7
, mark_0(2) -> 4
, mark_1(2) -> 7
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 7
, c_1(8) -> 4
, c_1(8) -> 7
, h_0(2) -> 2
, h_1(7) -> 4
, h_1(7) -> 7
, a__c^#_0(2) -> 1
, a__c^#_0(6) -> 5
, a__c^#_1(8) -> 14
, a__c^#_1(11) -> 10
, a__c^#_1(13) -> 12
, c_1_0() -> 1
, c_1_0() -> 5
, c_1_1() -> 10
, c_1_1() -> 12
, c_1_1() -> 14
, a__h^#_0(2) -> 1
, a__h^#_0(4) -> 3
, a__h^#_1(7) -> 9
, c_2_0(1) -> 1
, c_2_0(5) -> 3
, c_2_1(10) -> 1
, c_2_1(12) -> 3
, c_2_1(14) -> 9
, mark^#_0(2) -> 1
, c_5_0(3) -> 1
, c_5_1(9) -> 1}
6) { mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__h^#(X) -> c_2(a__c^#(d(X)))}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(X) -> c_2(a__c^#(d(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
and weakly orienting the rules
{ a__h^#(X) -> c_2(a__c^#(d(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [7]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [7]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [4]
c_1() = [0]
a__h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [13]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_2(a__c^#(d(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(7) -> 4
, a__f_1(7) -> 7
, f_0(2) -> 2
, f_1(7) -> 4
, f_1(7) -> 7
, f_1(15) -> 8
, a__c_0(2) -> 4
, a__c_1(2) -> 7
, a__c_1(8) -> 4
, a__c_1(8) -> 7
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 7
, g_1(7) -> 15
, d_0(2) -> 2
, d_0(2) -> 4
, d_0(4) -> 6
, d_1(2) -> 7
, d_1(2) -> 11
, d_1(4) -> 13
, d_1(7) -> 8
, d_1(8) -> 4
, d_1(8) -> 7
, a__h_1(7) -> 4
, a__h_1(7) -> 7
, mark_0(2) -> 4
, mark_1(2) -> 7
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 7
, c_1(8) -> 4
, c_1(8) -> 7
, h_0(2) -> 2
, h_1(7) -> 4
, h_1(7) -> 7
, a__c^#_0(2) -> 1
, a__c^#_0(6) -> 5
, a__c^#_1(8) -> 14
, a__c^#_1(11) -> 10
, a__c^#_1(13) -> 12
, a__h^#_0(2) -> 1
, a__h^#_0(4) -> 3
, a__h^#_1(7) -> 9
, c_2_0(1) -> 1
, c_2_0(5) -> 3
, c_2_1(10) -> 1
, c_2_1(12) -> 3
, c_2_1(14) -> 9
, mark^#_0(2) -> 1
, c_5_0(3) -> 1
, c_5_1(9) -> 1}
7) { mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [3]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f^#(X) -> c_8()}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(X) -> c_8()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [3]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ a__f^#(X) -> c_8()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [3]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [3]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, f_0(2) -> 2
, f_1(5) -> 4
, f_1(5) -> 5
, f_1(8) -> 6
, a__c_0(2) -> 4
, a__c_1(2) -> 5
, a__c_1(6) -> 4
, a__c_1(6) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 5
, g_1(5) -> 8
, d_0(2) -> 2
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(5) -> 6
, d_1(6) -> 4
, d_1(6) -> 5
, a__h_1(5) -> 4
, a__h_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 5
, c_1(6) -> 4
, c_1(6) -> 5
, h_0(2) -> 2
, h_1(5) -> 4
, h_1(5) -> 5
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 7
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(7) -> 1
, c_8_0() -> 1
, c_8_0() -> 3
, c_8_1() -> 7}
8) { mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_10()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_10()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [3]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__h^#(X) -> c_10()}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(X) -> c_10()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
and weakly orienting the rules
{ a__h^#(X) -> c_10()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [3]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [3]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_10()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_10()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_10()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_10()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, a__h^#(X) -> c_10()
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, f_0(2) -> 2
, f_1(5) -> 4
, f_1(5) -> 5
, f_1(8) -> 6
, a__c_0(2) -> 4
, a__c_1(2) -> 5
, a__c_1(6) -> 4
, a__c_1(6) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 5
, g_1(5) -> 8
, d_0(2) -> 2
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(5) -> 6
, d_1(6) -> 4
, d_1(6) -> 5
, a__h_1(5) -> 4
, a__h_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 5
, c_1(6) -> 4
, c_1(6) -> 5
, h_0(2) -> 2
, h_1(5) -> 4
, h_1(5) -> 5
, a__h^#_0(2) -> 1
, a__h^#_0(4) -> 3
, a__h^#_1(5) -> 7
, mark^#_0(2) -> 1
, c_5_0(3) -> 1
, c_5_1(7) -> 1
, c_10_0() -> 1
, c_10_0() -> 3
, c_10_1() -> 7}
9) {mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [11]
g(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [6]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [5]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, f_0(2) -> 2
, f_1(5) -> 4
, f_1(5) -> 5
, f_1(8) -> 6
, a__c_0(2) -> 4
, a__c_1(2) -> 5
, a__c_1(6) -> 4
, a__c_1(6) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 5
, g_1(5) -> 8
, d_0(2) -> 2
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(5) -> 6
, d_1(6) -> 4
, d_1(6) -> 5
, a__h_1(5) -> 4
, a__h_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 5
, c_1(6) -> 4
, c_1(6) -> 5
, h_0(2) -> 2
, h_1(5) -> 4
, h_1(5) -> 5
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 7
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(7) -> 1}
10)
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(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(c(X)) -> a__c(X)
, mark(h(X)) -> a__h(mark(X))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__f(f(X)) -> a__c(f(g(f(X))))
, a__c(X) -> d(X)
, a__h(X) -> a__c(d(X))
, a__f(X) -> f(X)
, a__c(X) -> c(X)
, a__h(X) -> h(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
and weakly orienting the rules
{ mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(h(X)) -> c_5(a__h^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [11]
g(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [6]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(c(X)) -> a__c(X)}
and weakly orienting the rules
{ mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(c(X)) -> a__c(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
and weakly orienting the rules
{ mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
and weakly orienting the rules
{ a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [5]
f(x1) = [1] x1 + [0]
a__c(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(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))
, mark(h(X)) -> a__h(mark(X))}
Weak Rules:
{ a__f(f(X)) -> a__c(f(g(f(X))))
, a__f(X) -> f(X)
, a__h(X) -> a__c(d(X))
, a__h(X) -> h(X)
, mark(c(X)) -> a__c(X)
, mark^#(h(X)) -> c_5(a__h^#(mark(X)))
, mark(g(X)) -> g(X)
, mark(d(X)) -> d(X)
, a__c(X) -> d(X)
, a__c(X) -> c(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, f_0(2) -> 2
, f_1(5) -> 4
, f_1(5) -> 5
, f_1(8) -> 6
, a__c_0(2) -> 4
, a__c_1(2) -> 5
, a__c_1(6) -> 4
, a__c_1(6) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_1(2) -> 5
, g_1(5) -> 8
, d_0(2) -> 2
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(5) -> 6
, d_1(6) -> 4
, d_1(6) -> 5
, a__h_1(5) -> 4
, a__h_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_1(2) -> 5
, c_1(6) -> 4
, c_1(6) -> 5
, h_0(2) -> 2
, h_1(5) -> 4
, h_1(5) -> 5
, a__h^#_0(2) -> 1
, a__h^#_0(4) -> 3
, a__h^#_1(5) -> 7
, mark^#_0(2) -> 1
, c_5_0(3) -> 1
, c_5_1(7) -> 1}
11)
{ mark^#(c(X)) -> c_4(a__c^#(X))
, a__c^#(X) -> c_1()}
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__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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: {a__c^#(X) -> c_1()}
Weak Rules: {mark^#(c(X)) -> c_4(a__c^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__c^#(X) -> c_1()}
and weakly orienting the rules
{mark^#(c(X)) -> c_4(a__c^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#(X) -> c_1()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [1]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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:
{ a__c^#(X) -> c_1()
, mark^#(c(X)) -> c_4(a__c^#(X))}
Details:
The given problem does not contain any strict rules
12)
{mark^#(c(X)) -> c_4(a__c^#(X))}
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__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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^#(c(X)) -> c_4(a__c^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(c(X)) -> c_4(a__c^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(c(X)) -> c_4(a__c^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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^#(c(X)) -> c_4(a__c^#(X))}
Details:
The given problem does not contain any strict rules
13)
{ mark^#(c(X)) -> c_4(a__c^#(X))
, a__c^#(X) -> c_9()}
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__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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: {a__c^#(X) -> c_9()}
Weak Rules: {mark^#(c(X)) -> c_4(a__c^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__c^#(X) -> c_9()}
and weakly orienting the rules
{mark^#(c(X)) -> c_4(a__c^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#(X) -> c_9()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [1] x1 + [1]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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:
{ a__c^#(X) -> c_9()
, mark^#(c(X)) -> c_4(a__c^#(X))}
Details:
The given problem does not contain any strict rules
14)
{mark^#(d(X)) -> c_7()}
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__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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^#(d(X)) -> c_7()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(d(X)) -> c_7()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(d(X)) -> c_7()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [1] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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^#(d(X)) -> c_7()}
Details:
The given problem does not contain any strict rules
15)
{mark^#(g(X)) -> c_6()}
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__c(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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_6()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(g(X)) -> c_6()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(g(X)) -> c_6()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__c^#(x1) = [0] x1 + [0]
c_1() = [0]
a__h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
mark^#(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() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [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_6()}
Details:
The given problem does not contain any strict rules