'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(ok(X)) -> ok(f(X))
, g(ok(X)) -> ok(g(X))
, h(ok(X)) -> ok(h(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(X)) -> c_0(g^#(h(f(X))))
, active^#(f(X)) -> c_1(f^#(active(X)))
, active^#(h(X)) -> c_2(h^#(active(X)))
, f^#(mark(X)) -> c_3(f^#(X))
, h^#(mark(X)) -> c_4(h^#(X))
, proper^#(f(X)) -> c_5(f^#(proper(X)))
, proper^#(g(X)) -> c_6(g^#(proper(X)))
, proper^#(h(X)) -> c_7(h^#(proper(X)))
, f^#(ok(X)) -> c_8(f^#(X))
, g^#(ok(X)) -> c_9(g^#(X))
, h^#(ok(X)) -> c_10(h^#(X))
, top^#(mark(X)) -> c_11(top^#(proper(X)))
, top^#(ok(X)) -> c_12(top^#(active(X)))}
The usable rules are:
{ active(f(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
The estimated dependency graph contains the following edges:
{active^#(f(X)) -> c_0(g^#(h(f(X))))}
==> {g^#(ok(X)) -> c_9(g^#(X))}
{active^#(f(X)) -> c_1(f^#(active(X)))}
==> {f^#(ok(X)) -> c_8(f^#(X))}
{active^#(f(X)) -> c_1(f^#(active(X)))}
==> {f^#(mark(X)) -> c_3(f^#(X))}
{active^#(h(X)) -> c_2(h^#(active(X)))}
==> {h^#(ok(X)) -> c_10(h^#(X))}
{active^#(h(X)) -> c_2(h^#(active(X)))}
==> {h^#(mark(X)) -> c_4(h^#(X))}
{f^#(mark(X)) -> c_3(f^#(X))}
==> {f^#(ok(X)) -> c_8(f^#(X))}
{f^#(mark(X)) -> c_3(f^#(X))}
==> {f^#(mark(X)) -> c_3(f^#(X))}
{h^#(mark(X)) -> c_4(h^#(X))}
==> {h^#(ok(X)) -> c_10(h^#(X))}
{h^#(mark(X)) -> c_4(h^#(X))}
==> {h^#(mark(X)) -> c_4(h^#(X))}
{proper^#(f(X)) -> c_5(f^#(proper(X)))}
==> {f^#(ok(X)) -> c_8(f^#(X))}
{proper^#(f(X)) -> c_5(f^#(proper(X)))}
==> {f^#(mark(X)) -> c_3(f^#(X))}
{proper^#(g(X)) -> c_6(g^#(proper(X)))}
==> {g^#(ok(X)) -> c_9(g^#(X))}
{proper^#(h(X)) -> c_7(h^#(proper(X)))}
==> {h^#(ok(X)) -> c_10(h^#(X))}
{proper^#(h(X)) -> c_7(h^#(proper(X)))}
==> {h^#(mark(X)) -> c_4(h^#(X))}
{f^#(ok(X)) -> c_8(f^#(X))}
==> {f^#(ok(X)) -> c_8(f^#(X))}
{f^#(ok(X)) -> c_8(f^#(X))}
==> {f^#(mark(X)) -> c_3(f^#(X))}
{g^#(ok(X)) -> c_9(g^#(X))}
==> {g^#(ok(X)) -> c_9(g^#(X))}
{h^#(ok(X)) -> c_10(h^#(X))}
==> {h^#(ok(X)) -> c_10(h^#(X))}
{h^#(ok(X)) -> c_10(h^#(X))}
==> {h^#(mark(X)) -> c_4(h^#(X))}
{top^#(mark(X)) -> c_11(top^#(proper(X)))}
==> {top^#(ok(X)) -> c_12(top^#(active(X)))}
{top^#(mark(X)) -> c_11(top^#(proper(X)))}
==> {top^#(mark(X)) -> c_11(top^#(proper(X)))}
{top^#(ok(X)) -> c_12(top^#(active(X)))}
==> {top^#(ok(X)) -> c_12(top^#(active(X)))}
{top^#(ok(X)) -> c_12(top^#(active(X)))}
==> {top^#(mark(X)) -> c_11(top^#(proper(X)))}
We consider the following path(s):
1) { top^#(mark(X)) -> c_11(top^#(proper(X)))
, top^#(ok(X)) -> c_12(top^#(active(X)))}
The usable rules for this path are the following:
{ active(f(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, top^#(mark(X)) -> c_11(top^#(proper(X)))
, top^#(ok(X)) -> c_12(top^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{top^#(ok(X)) -> c_12(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_12(top^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_11(x1) = [1] x1 + [7]
c_12(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(f(X)) -> mark(g(h(f(X))))}
and weakly orienting the rules
{top^#(ok(X)) -> c_12(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(f(X)) -> mark(g(h(f(X))))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [0]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{top^#(mark(X)) -> c_11(top^#(proper(X)))}
and weakly orienting the rules
{ active(f(X)) -> mark(g(h(f(X))))
, top^#(ok(X)) -> c_12(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{top^#(mark(X)) -> c_11(top^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [8]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [2]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [0]
ok(x1) = [1] x1 + [9]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [7]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ top^#(mark(X)) -> c_11(top^#(proper(X)))
, active(f(X)) -> mark(g(h(f(X))))
, top^#(ok(X)) -> c_12(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))
, active(h(X)) -> h(active(X))
, proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ top^#(mark(X)) -> c_11(top^#(proper(X)))
, active(f(X)) -> mark(g(h(f(X))))
, top^#(ok(X)) -> c_12(top^#(active(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ active_0(3) -> 29
, active_0(7) -> 29
, mark_0(3) -> 3
, mark_0(7) -> 3
, proper_0(3) -> 27
, proper_0(7) -> 27
, ok_0(3) -> 7
, ok_0(7) -> 7
, top^#_0(3) -> 25
, top^#_0(7) -> 25
, top^#_0(27) -> 26
, top^#_0(29) -> 28
, c_11_0(26) -> 25
, c_12_0(28) -> 25}
2) { active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(ok(X)) -> c_8(f^#(X))
, f^#(mark(X)) -> c_3(f^#(X))}
The usable rules for this path are the following:
{ active(f(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(ok(X)) -> c_8(f^#(X))
, f^#(mark(X)) -> c_3(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(ok(X)) -> c_8(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_8(f^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [1]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(mark(X)) -> c_3(f^#(X))}
and weakly orienting the rules
{f^#(ok(X)) -> c_8(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(mark(X)) -> c_3(f^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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^#(mark(X)) -> c_3(f^#(X))
, f^#(ok(X)) -> c_8(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]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [10]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(f(X)) -> mark(g(h(f(X))))}
and weakly orienting the rules
{ active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(mark(X)) -> c_3(f^#(X))
, f^#(ok(X)) -> c_8(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(f(X)) -> mark(g(h(f(X))))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(mark(X)) -> c_3(f^#(X))
, f^#(ok(X)) -> c_8(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))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, active^#(f(X)) -> c_1(f^#(active(X)))
, f^#(mark(X)) -> c_3(f^#(X))
, f^#(ok(X)) -> c_8(f^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, active^#_0(3) -> 9
, active^#_0(7) -> 9
, f^#_0(3) -> 13
, f^#_0(7) -> 13
, c_3_0(13) -> 13
, c_8_0(13) -> 13}
3) { active^#(h(X)) -> c_2(h^#(active(X)))
, h^#(ok(X)) -> c_10(h^#(X))
, h^#(mark(X)) -> c_4(h^#(X))}
The usable rules for this path are the following:
{ active(f(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, active^#(h(X)) -> c_2(h^#(active(X)))
, h^#(ok(X)) -> c_10(h^#(X))
, h^#(mark(X)) -> c_4(h^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{h^#(ok(X)) -> c_10(h^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(ok(X)) -> c_10(h^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [9]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h^#(mark(X)) -> c_4(h^#(X))}
and weakly orienting the rules
{h^#(ok(X)) -> c_10(h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(mark(X)) -> c_4(h^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(h(X)) -> c_2(h^#(active(X)))}
and weakly orienting the rules
{ h^#(mark(X)) -> c_4(h^#(X))
, h^#(ok(X)) -> c_10(h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(h(X)) -> c_2(h^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(f(X)) -> mark(g(h(f(X))))}
and weakly orienting the rules
{ active^#(h(X)) -> c_2(h^#(active(X)))
, h^#(mark(X)) -> c_4(h^#(X))
, h^#(ok(X)) -> c_10(h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(f(X)) -> mark(g(h(f(X))))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [3]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, active^#(h(X)) -> c_2(h^#(active(X)))
, h^#(mark(X)) -> c_4(h^#(X))
, h^#(ok(X)) -> c_10(h^#(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))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, active^#(h(X)) -> c_2(h^#(active(X)))
, h^#(mark(X)) -> c_4(h^#(X))
, h^#(ok(X)) -> c_10(h^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(2) -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, h^#_0(2) -> 1
, c_4_0(1) -> 1
, c_10_0(1) -> 1}
4) { proper^#(f(X)) -> c_5(f^#(proper(X)))
, f^#(ok(X)) -> c_8(f^#(X))
, f^#(mark(X)) -> c_3(f^#(X))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, proper^#(f(X)) -> c_5(f^#(proper(X)))
, f^#(ok(X)) -> c_8(f^#(X))
, f^#(mark(X)) -> c_3(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(ok(X)) -> c_8(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_8(f^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [1]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(f(X)) -> c_5(f^#(proper(X)))}
and weakly orienting the rules
{f^#(ok(X)) -> c_8(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(f(X)) -> c_5(f^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_5(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(mark(X)) -> c_3(f^#(X))}
and weakly orienting the rules
{ proper^#(f(X)) -> c_5(f^#(proper(X)))
, f^#(ok(X)) -> c_8(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(mark(X)) -> c_3(f^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [13]
c_5(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ f^#(mark(X)) -> c_3(f^#(X))
, proper^#(f(X)) -> c_5(f^#(proper(X)))
, f^#(ok(X)) -> c_8(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ f^#(mark(X)) -> c_3(f^#(X))
, proper^#(f(X)) -> c_5(f^#(proper(X)))
, f^#(ok(X)) -> c_8(f^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, f^#_0(3) -> 13
, f^#_0(7) -> 13
, c_3_0(13) -> 13
, proper^#_0(3) -> 18
, proper^#_0(7) -> 18
, c_8_0(13) -> 13}
5) { proper^#(h(X)) -> c_7(h^#(proper(X)))
, h^#(ok(X)) -> c_10(h^#(X))
, h^#(mark(X)) -> c_4(h^#(X))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, proper^#(h(X)) -> c_7(h^#(proper(X)))
, h^#(ok(X)) -> c_10(h^#(X))
, h^#(mark(X)) -> c_4(h^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{h^#(ok(X)) -> c_10(h^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(ok(X)) -> c_10(h^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(h(X)) -> c_7(h^#(proper(X)))}
and weakly orienting the rules
{h^#(ok(X)) -> c_10(h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(h(X)) -> c_7(h^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [9]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h^#(mark(X)) -> c_4(h^#(X))}
and weakly orienting the rules
{ proper^#(h(X)) -> c_7(h^#(proper(X)))
, h^#(ok(X)) -> c_10(h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(mark(X)) -> c_4(h^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ h^#(mark(X)) -> c_4(h^#(X))
, proper^#(h(X)) -> c_7(h^#(proper(X)))
, h^#(ok(X)) -> c_10(h^#(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ h^#(mark(X)) -> c_4(h^#(X))
, proper^#(h(X)) -> c_7(h^#(proper(X)))
, h^#(ok(X)) -> c_10(h^#(X))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, h^#_0(3) -> 15
, h^#_0(7) -> 15
, c_4_0(15) -> 15
, proper^#_0(3) -> 18
, proper^#_0(7) -> 18
, c_10_0(15) -> 15}
6) {active^#(h(X)) -> c_2(h^#(active(X)))}
The usable rules for this path are the following:
{ active(f(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, active^#(h(X)) -> c_2(h^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(h(X)) -> c_2(h^#(active(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(h(X)) -> c_2(h^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(f(X)) -> mark(g(h(f(X))))}
and weakly orienting the rules
{active^#(h(X)) -> c_2(h^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(f(X)) -> mark(g(h(f(X))))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [5]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [11]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, active^#(h(X)) -> c_2(h^#(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))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, active^#(h(X)) -> c_2(h^#(active(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, active^#_0(3) -> 9
, active^#_0(7) -> 9
, h^#_0(3) -> 15
, h^#_0(7) -> 15}
7) { proper^#(g(X)) -> c_6(g^#(proper(X)))
, g^#(ok(X)) -> c_9(g^#(X))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, proper^#(g(X)) -> c_6(g^#(proper(X)))
, g^#(ok(X)) -> c_9(g^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(g(X)) -> c_6(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_6(g^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [1] x1 + [3]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [6]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(ok(X)) -> c_9(g^#(X))}
and weakly orienting the rules
{proper^#(g(X)) -> c_6(g^#(proper(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(ok(X)) -> c_9(g^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ g^#(ok(X)) -> c_9(g^#(X))
, proper^#(g(X)) -> c_6(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ g^#(ok(X)) -> c_9(g^#(X))
, proper^#(g(X)) -> c_6(g^#(proper(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, g^#_0(3) -> 11
, g^#_0(7) -> 11
, proper^#_0(3) -> 18
, proper^#_0(7) -> 18
, c_9_0(11) -> 11}
8) {active^#(f(X)) -> c_1(f^#(active(X)))}
The usable rules for this path are the following:
{ active(f(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(X)) -> mark(g(h(f(X))))
, active(f(X)) -> f(active(X))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(f(X)) -> mark(g(h(f(X))))}
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(X)) -> mark(g(h(f(X))))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [7]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [15]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [4]
f^#(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, 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))
, active(h(X)) -> h(active(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules:
{ active(f(X)) -> mark(g(h(f(X))))
, 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:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, active^#_0(3) -> 9
, active^#_0(7) -> 9
, f^#_0(3) -> 13
, f^#_0(7) -> 13}
9) {proper^#(g(X)) -> c_6(g^#(proper(X)))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, proper^#(g(X)) -> c_6(g^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(g(X)) -> c_6(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_6(g^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules: {proper^#(g(X)) -> c_6(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules: {proper^#(g(X)) -> c_6(g^#(proper(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, g^#_0(3) -> 11
, g^#_0(7) -> 11
, proper^#_0(3) -> 18
, proper^#_0(7) -> 18}
10)
{proper^#(h(X)) -> c_7(h^#(proper(X)))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, proper^#(h(X)) -> c_7(h^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(h(X)) -> c_7(h^#(proper(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(h(X)) -> c_7(h^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules: {proper^#(h(X)) -> c_7(h^#(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules: {proper^#(h(X)) -> c_7(h^#(proper(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, h^#_0(3) -> 15
, h^#_0(7) -> 15
, proper^#_0(3) -> 18
, proper^#_0(7) -> 18}
11)
{proper^#(f(X)) -> c_5(f^#(proper(X)))}
The usable rules for this path are the following:
{ proper(f(X)) -> f(proper(X))
, proper(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(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(g(X)) -> g(proper(X))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))
, proper^#(f(X)) -> c_5(f^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(f(X)) -> c_5(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_5(f^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
h(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]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [7]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules: {proper^#(f(X)) -> c_5(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))
, proper(h(X)) -> h(proper(X))
, f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, g(ok(X)) -> ok(g(X))}
Weak Rules: {proper^#(f(X)) -> c_5(f^#(proper(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, f^#_0(3) -> 13
, f^#_0(7) -> 13
, proper^#_0(3) -> 18
, proper^#_0(7) -> 18}
12)
{ active^#(f(X)) -> c_0(g^#(h(f(X))))
, g^#(ok(X)) -> c_9(g^#(X))}
The usable rules for this path are the following:
{ f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(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:
{ f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, active^#(f(X)) -> c_0(g^#(h(f(X))))
, g^#(ok(X)) -> c_9(g^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(f(X)) -> c_0(g^#(h(f(X))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(f(X)) -> c_0(g^#(h(f(X))))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [1]
mark(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [1]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(ok(X)) -> c_9(g^#(X))}
and weakly orienting the rules
{active^#(f(X)) -> c_0(g^#(h(f(X))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(ok(X)) -> c_9(g^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [1]
mark(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [1]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))}
Weak Rules:
{ g^#(ok(X)) -> c_9(g^#(X))
, active^#(f(X)) -> c_0(g^#(h(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:
{ f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))}
Weak Rules:
{ g^#(ok(X)) -> c_9(g^#(X))
, active^#(f(X)) -> c_0(g^#(h(f(X))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, active^#_0(3) -> 9
, active^#_0(7) -> 9
, g^#_0(3) -> 11
, g^#_0(7) -> 11
, c_9_0(11) -> 11}
13)
{active^#(f(X)) -> c_0(g^#(h(f(X))))}
The usable rules for this path are the following:
{ f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(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:
{ f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))
, active^#(f(X)) -> c_0(g^#(h(f(X))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(f(X)) -> c_0(g^#(h(f(X))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(f(X)) -> c_0(g^#(h(f(X))))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [1]
mark(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [1]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))}
Weak Rules: {active^#(f(X)) -> c_0(g^#(h(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:
{ f(mark(X)) -> mark(f(X))
, h(mark(X)) -> mark(h(X))
, f(ok(X)) -> ok(f(X))
, h(ok(X)) -> ok(h(X))}
Weak Rules: {active^#(f(X)) -> c_0(g^#(h(f(X))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(7) -> 3
, ok_0(3) -> 7
, ok_0(7) -> 7
, active^#_0(3) -> 9
, active^#_0(7) -> 9
, g^#_0(3) -> 11
, g^#_0(7) -> 11}