'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, top(mark(X)) -> top(proper(X))
, top(ok(X)) -> top(active(X))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ active^#(f(f(a()))) -> c_0(f^#(g(f(a()))))
, active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(mark(X)) -> c_2(f^#(X))
, proper^#(f(X)) -> c_3(f^#(proper(X)))
, proper^#(a()) -> c_4()
, proper^#(g(X)) -> c_5(g^#(proper(X)))
, f^#(ok(X)) -> c_6(f^#(X))
, g^#(ok(X)) -> c_7(g^#(X))
, top^#(mark(X)) -> c_8(top^#(proper(X)))
, top^#(ok(X)) -> c_9(top^#(active(X)))}
The usable rules are:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
The estimated dependency graph contains the following edges:
{active^#(f(X)) -> c_1(f^#(active(X)))}
==> {f^#(ok(X)) -> c_6(f^#(X))}
{active^#(f(X)) -> c_1(f^#(active(X)))}
==> {f^#(mark(X)) -> c_2(f^#(X))}
{f^#(mark(X)) -> c_2(f^#(X))}
==> {f^#(ok(X)) -> c_6(f^#(X))}
{f^#(mark(X)) -> c_2(f^#(X))}
==> {f^#(mark(X)) -> c_2(f^#(X))}
{proper^#(f(X)) -> c_3(f^#(proper(X)))}
==> {f^#(ok(X)) -> c_6(f^#(X))}
{proper^#(f(X)) -> c_3(f^#(proper(X)))}
==> {f^#(mark(X)) -> c_2(f^#(X))}
{proper^#(g(X)) -> c_5(g^#(proper(X)))}
==> {g^#(ok(X)) -> c_7(g^#(X))}
{f^#(ok(X)) -> c_6(f^#(X))}
==> {f^#(ok(X)) -> c_6(f^#(X))}
{f^#(ok(X)) -> c_6(f^#(X))}
==> {f^#(mark(X)) -> c_2(f^#(X))}
{g^#(ok(X)) -> c_7(g^#(X))}
==> {g^#(ok(X)) -> c_7(g^#(X))}
{top^#(mark(X)) -> c_8(top^#(proper(X)))}
==> {top^#(ok(X)) -> c_9(top^#(active(X)))}
{top^#(mark(X)) -> c_8(top^#(proper(X)))}
==> {top^#(mark(X)) -> c_8(top^#(proper(X)))}
{top^#(ok(X)) -> c_9(top^#(active(X)))}
==> {top^#(ok(X)) -> c_9(top^#(active(X)))}
{top^#(ok(X)) -> c_9(top^#(active(X)))}
==> {top^#(mark(X)) -> c_8(top^#(proper(X)))}
We consider the following path(s):
1) { top^#(mark(X)) -> c_8(top^#(proper(X)))
, top^#(ok(X)) -> c_9(top^#(active(X)))}
The usable rules for this path are the following:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active(f(X)) -> f(active(X))
, proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(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:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active(f(X)) -> f(active(X))
, proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, top^#(mark(X)) -> c_8(top^#(proper(X)))
, top^#(ok(X)) -> c_9(top^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{top^#(ok(X)) -> c_9(top^#(active(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{top^#(ok(X)) -> c_9(top^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [5]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_8(x1) = [1] x1 + [5]
c_9(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper(a()) -> ok(a())}
and weakly orienting the rules
{top^#(ok(X)) -> c_9(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper(a()) -> ok(a())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [5]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [3]
c_8(x1) = [1] x1 + [4]
c_9(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(f(f(a()))) -> mark(f(g(f(a()))))}
and weakly orienting the rules
{ proper(a()) -> ok(a())
, top^#(ok(X)) -> c_9(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(f(f(a()))) -> mark(f(g(f(a()))))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [8]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [12]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [2]
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:
{ active(f(X)) -> f(active(X))
, proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, top^#(mark(X)) -> c_8(top^#(proper(X)))}
Weak Rules:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, proper(a()) -> ok(a())
, top^#(ok(X)) -> c_9(top^#(active(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:
{ active(f(X)) -> f(active(X))
, proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, top^#(mark(X)) -> c_8(top^#(proper(X)))}
Weak Rules:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, proper(a()) -> ok(a())
, top^#(ok(X)) -> c_9(top^#(active(X)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ active_0(2) -> 4
, active_1(2) -> 9
, active_1(7) -> 11
, a_0() -> 2
, a_1() -> 7
, mark_0(2) -> 2
, proper_1(2) -> 6
, ok_0(2) -> 2
, ok_1(7) -> 6
, top^#_0(2) -> 1
, top^#_0(4) -> 3
, top^#_1(6) -> 5
, top^#_1(9) -> 8
, top^#_1(11) -> 10
, c_8_1(5) -> 1
, c_9_0(3) -> 1
, c_9_1(8) -> 1
, c_9_1(10) -> 5}
2) { active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(ok(X)) -> c_6(f^#(X))
, f^#(mark(X)) -> c_2(f^#(X))}
The usable rules for this path are the following:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(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:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(ok(X)) -> c_6(f^#(X))
, f^#(mark(X)) -> c_2(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(mark(X)) -> c_2(f^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(mark(X)) -> c_2(f^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [6]
c_2(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [7]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(ok(X)) -> c_6(f^#(X))}
and weakly orienting the rules
{f^#(mark(X)) -> c_2(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(ok(X)) -> c_6(f^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(f(X)) -> c_1(f^#(active(X)))}
and weakly orienting the rules
{ f^#(ok(X)) -> c_6(f^#(X))
, f^#(mark(X)) -> c_2(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(f(X)) -> c_1(f^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(f(f(a()))) -> mark(f(g(f(a()))))}
and weakly orienting the rules
{ active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(ok(X)) -> c_6(f^#(X))
, f^#(mark(X)) -> c_2(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(f(f(a()))) -> mark(f(g(f(a()))))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(ok(X)) -> c_6(f^#(X))
, f^#(mark(X)) -> c_2(f^#(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:
{ active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(ok(X)) -> c_6(f^#(X))
, f^#(mark(X)) -> c_2(f^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(7) -> 4
, ok_0(3) -> 7
, ok_0(4) -> 7
, ok_0(7) -> 7
, active^#_0(3) -> 9
, active^#_0(4) -> 9
, active^#_0(7) -> 9
, f^#_0(3) -> 11
, f^#_0(4) -> 11
, f^#_0(7) -> 11
, c_2_0(11) -> 11
, c_6_0(11) -> 11}
3) { proper^#(f(X)) -> c_3(f^#(proper(X)))
, f^#(ok(X)) -> c_6(f^#(X))
, f^#(mark(X)) -> c_2(f^#(X))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(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:
{ proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, proper^#(f(X)) -> c_3(f^#(proper(X)))
, f^#(ok(X)) -> c_6(f^#(X))
, f^#(mark(X)) -> c_2(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(ok(X)) -> c_6(f^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(ok(X)) -> c_6(f^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(f(X)) -> c_3(f^#(proper(X)))}
and weakly orienting the rules
{f^#(ok(X)) -> c_6(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(f(X)) -> c_3(f^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [4]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(mark(X)) -> c_2(f^#(X))}
and weakly orienting the rules
{ proper^#(f(X)) -> c_3(f^#(proper(X)))
, f^#(ok(X)) -> c_6(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(mark(X)) -> c_2(f^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [12]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper(a()) -> ok(a())}
and weakly orienting the rules
{ f^#(mark(X)) -> c_2(f^#(X))
, proper^#(f(X)) -> c_3(f^#(proper(X)))
, f^#(ok(X)) -> c_6(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper(a()) -> ok(a())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [5]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [3]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ proper(a()) -> ok(a())
, f^#(mark(X)) -> c_2(f^#(X))
, proper^#(f(X)) -> c_3(f^#(proper(X)))
, f^#(ok(X)) -> c_6(f^#(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:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ proper(a()) -> ok(a())
, f^#(mark(X)) -> c_2(f^#(X))
, proper^#(f(X)) -> c_3(f^#(proper(X)))
, f^#(ok(X)) -> c_6(f^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(7) -> 4
, ok_0(3) -> 7
, ok_0(4) -> 7
, ok_0(7) -> 7
, f^#_0(3) -> 11
, f^#_0(4) -> 11
, f^#_0(7) -> 11
, c_2_0(11) -> 11
, proper^#_0(3) -> 14
, proper^#_0(4) -> 14
, proper^#_0(7) -> 14
, c_6_0(11) -> 11}
4) {active^#(f(X)) -> c_1(f^#(active(X)))}
The usable rules for this path are the following:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(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:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, active^#(f(X)) -> c_1(f^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(f(X)) -> c_1(f^#(active(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(f(X)) -> c_1(f^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [3]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(f(f(a()))) -> mark(f(g(f(a()))))}
and weakly orienting the rules
{active^#(f(X)) -> c_1(f^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(f(f(a()))) -> mark(f(g(f(a()))))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active^#(f(X)) -> c_1(f^#(active(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:
{ active(f(X)) -> f(active(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(f(a()))) -> mark(f(g(f(a()))))
, active^#(f(X)) -> c_1(f^#(active(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(7) -> 4
, ok_0(3) -> 7
, ok_0(4) -> 7
, ok_0(7) -> 7
, active^#_0(3) -> 9
, active^#_0(4) -> 9
, active^#_0(7) -> 9
, f^#_0(3) -> 11
, f^#_0(4) -> 11
, f^#_0(7) -> 11}
5) { proper^#(g(X)) -> c_5(g^#(proper(X)))
, g^#(ok(X)) -> c_7(g^#(X))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(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:
{ proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, proper^#(g(X)) -> c_5(g^#(proper(X)))
, g^#(ok(X)) -> c_7(g^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ proper^#(g(X)) -> c_5(g^#(proper(X)))
, g^#(ok(X)) -> c_7(g^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper^#(g(X)) -> c_5(g^#(proper(X)))
, g^#(ok(X)) -> c_7(g^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [8]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
g^#(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper(a()) -> ok(a())}
and weakly orienting the rules
{ proper^#(g(X)) -> c_5(g^#(proper(X)))
, g^#(ok(X)) -> c_7(g^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper(a()) -> ok(a())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [4]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [13]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [9]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ proper(a()) -> ok(a())
, proper^#(g(X)) -> c_5(g^#(proper(X)))
, g^#(ok(X)) -> c_7(g^#(X))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ proper(a()) -> ok(a())
, proper^#(g(X)) -> c_5(g^#(proper(X)))
, g^#(ok(X)) -> c_7(g^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(7) -> 4
, ok_0(3) -> 7
, ok_0(4) -> 7
, ok_0(7) -> 7
, proper^#_0(3) -> 14
, proper^#_0(4) -> 14
, proper^#_0(7) -> 14
, g^#_0(3) -> 18
, g^#_0(4) -> 18
, g^#_0(7) -> 18
, c_7_0(18) -> 18}
6) {proper^#(f(X)) -> c_3(f^#(proper(X)))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(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:
{ proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, proper^#(f(X)) -> c_3(f^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(f(X)) -> c_3(f^#(proper(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(f(X)) -> c_3(f^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper(a()) -> ok(a())}
and weakly orienting the rules
{proper^#(f(X)) -> c_3(f^#(proper(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper(a()) -> ok(a())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [4]
mark(x1) = [1] x1 + [2]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [4]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [1]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ proper(a()) -> ok(a())
, proper^#(f(X)) -> c_3(f^#(proper(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:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ proper(a()) -> ok(a())
, proper^#(f(X)) -> c_3(f^#(proper(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(7) -> 4
, ok_0(3) -> 7
, ok_0(4) -> 7
, ok_0(7) -> 7
, f^#_0(3) -> 11
, f^#_0(4) -> 11
, f^#_0(7) -> 11
, proper^#_0(3) -> 14
, proper^#_0(4) -> 14
, proper^#_0(7) -> 14}
7) {proper^#(g(X)) -> c_5(g^#(proper(X)))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(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:
{ proper(f(X)) -> f(proper(X))
, proper(a()) -> ok(a())
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, proper^#(g(X)) -> c_5(g^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(g(X)) -> c_5(g^#(proper(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(g(X)) -> c_5(g^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper(a()) -> ok(a())}
and weakly orienting the rules
{proper^#(g(X)) -> c_5(g^#(proper(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper(a()) -> ok(a())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [5]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [11]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [5]
g^#(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ proper(a()) -> ok(a())
, proper^#(g(X)) -> c_5(g^#(proper(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:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ proper(a()) -> ok(a())
, proper^#(g(X)) -> c_5(g^#(proper(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(7) -> 4
, ok_0(3) -> 7
, ok_0(4) -> 7
, ok_0(7) -> 7
, proper^#_0(3) -> 14
, proper^#_0(4) -> 14
, proper^#_0(7) -> 14
, g^#_0(3) -> 18
, g^#_0(4) -> 18
, g^#_0(7) -> 18}
8) {active^#(f(f(a()))) -> c_0(f^#(g(f(a()))))}
The usable rules for this path are the following:
{ f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(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:
{ f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, active^#(f(f(a()))) -> c_0(f^#(g(f(a()))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(f(f(a()))) -> c_0(f^#(g(f(a()))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(f(f(a()))) -> c_0(f^#(g(f(a()))))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [1]
a() = [14]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [2]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules: {active^#(f(f(a()))) -> c_0(f^#(g(f(a()))))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ f(mark(X)) -> mark(f(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules: {active^#(f(f(a()))) -> c_0(f^#(g(f(a()))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0() -> 3
, mark_0(3) -> 4
, mark_0(4) -> 4
, mark_0(7) -> 4
, ok_0(3) -> 7
, ok_0(4) -> 7
, ok_0(7) -> 7
, active^#_0(3) -> 9
, active^#_0(4) -> 9
, active^#_0(7) -> 9
, f^#_0(3) -> 11
, f^#_0(4) -> 11
, f^#_0(7) -> 11}
9) {proper^#(a()) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {proper^#(a()) -> c_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(a()) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(a()) -> c_4()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {proper^#(a()) -> c_4()}
Details:
The given problem does not contain any strict rules