'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ merge(x, nil()) -> x
, merge(nil(), y) -> y
, merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
, merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v()))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ merge^#(x, nil()) -> c_0()
, merge^#(nil(), y) -> c_1()
, merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))
, merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(++(x, y), v()))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
==> {merge^#(++(x, y), ++(u(), v())) ->
c_3(merge^#(++(x, y), v()))}
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
==> {merge^#(++(x, y), ++(u(), v())) ->
c_2(merge^#(y, ++(u(), v())))}
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
==> {merge^#(nil(), y) -> c_1()}
We consider the following path(s):
1) { merge^#(++(x, y), ++(u(), v())) ->
c_2(merge^#(y, ++(u(), v())))
, merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(++(x, y), v()))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
merge(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
u() = [0]
v() = [0]
merge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(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:
{merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(++(x, y), v()))}
Weak Rules:
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
Details:
We apply the weight gap principle, strictly orienting the rules
{merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(++(x, y), v()))}
and weakly orienting the rules
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(++(x, y), v()))}
Details:
Interpretation Functions:
merge(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
++(x1, x2) = [1] x1 + [1] x2 + [8]
u() = [0]
v() = [0]
merge^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [8]
c_3(x1) = [1] 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:
{ merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(++(x, y), v()))
, merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
Details:
The given problem does not contain any strict rules
2) { merge^#(++(x, y), ++(u(), v())) ->
c_2(merge^#(y, ++(u(), v())))
, merge^#(nil(), y) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
merge(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
u() = [0]
v() = [0]
merge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(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: {merge^#(nil(), y) -> c_1()}
Weak Rules:
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
Details:
We apply the weight gap principle, strictly orienting the rules
{merge^#(nil(), y) -> c_1()}
and weakly orienting the rules
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{merge^#(nil(), y) -> c_1()}
Details:
Interpretation Functions:
merge(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
++(x1, x2) = [1] x1 + [1] x2 + [0]
u() = [0]
v() = [0]
merge^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3(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:
{ merge^#(nil(), y) -> c_1()
, merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
Details:
The given problem does not contain any strict rules
3) {merge^#(++(x, y), ++(u(), v())) ->
c_2(merge^#(y, ++(u(), v())))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
merge(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
u() = [0]
v() = [0]
merge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(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:
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
Details:
Interpretation Functions:
merge(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
++(x1, x2) = [1] x1 + [1] x2 + [8]
u() = [0]
v() = [0]
merge^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [2]
c_3(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:
{merge^#(++(x, y), ++(u(), v())) -> c_2(merge^#(y, ++(u(), v())))}
Details:
The given problem does not contain any strict rules
4) {merge^#(x, nil()) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
merge(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
u() = [0]
v() = [0]
merge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(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: {merge^#(x, nil()) -> c_0()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{merge^#(x, nil()) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{merge^#(x, nil()) -> c_0()}
Details:
Interpretation Functions:
merge(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
u() = [0]
v() = [0]
merge^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(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: {merge^#(x, nil()) -> c_0()}
Details:
The given problem does not contain any strict rules