'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +(s(x), s(y)) -> s(+(s(x), +(y, 0())))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ +^#(0(), y) -> c_0()
, +^#(s(x), 0()) -> c_1()
, +^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
The usable rules are:
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +(s(x), s(y)) -> s(+(s(x), +(y, 0())))}
The estimated dependency graph contains the following edges:
{+^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
==> {+^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
{+^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
==> {+^#(s(x), 0()) -> c_1()}
We consider the following path(s):
1) { +^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))
, +^#(s(x), 0()) -> c_1()}
The usable rules for this path are the following:
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +(s(x), s(y)) -> s(+(s(x), +(y, 0())))}
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:
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +(s(x), s(y)) -> s(+(s(x), +(y, 0())))
, +^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))
, +^#(s(x), 0()) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +^#(s(x), 0()) -> c_1()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +^#(s(x), 0()) -> c_1()}
Details:
Interpretation Functions:
+(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
+^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{+^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
and weakly orienting the rules
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +^#(s(x), 0()) -> c_1()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{+^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
Details:
Interpretation Functions:
+(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [1]
+^#(x1, x2) = [1] x1 + [1] x2 + [14]
c_0() = [0]
c_1() = [0]
c_2(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: {+(s(x), s(y)) -> s(+(s(x), +(y, 0())))}
Weak Rules:
{ +^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))
, +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +^#(s(x), 0()) -> c_1()}
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: {+(s(x), s(y)) -> s(+(s(x), +(y, 0())))}
Weak Rules:
{ +^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))
, +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +^#(s(x), 0()) -> c_1()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ +_0(2, 2) -> 6
, +_0(3, 2) -> 6
, 0_0() -> 2
, 0_0() -> 6
, s_0(2) -> 3
, s_0(2) -> 6
, s_0(3) -> 3
, s_0(3) -> 6
, +^#_0(2, 2) -> 4
, +^#_0(2, 3) -> 4
, +^#_0(3, 2) -> 4
, +^#_0(3, 3) -> 4
, +^#_0(3, 6) -> 5
, c_1_0() -> 4
, c_1_0() -> 5
, c_2_0(5) -> 4
, c_2_0(5) -> 5}
2) {+^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
The usable rules for this path are the following:
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +(s(x), s(y)) -> s(+(s(x), +(y, 0())))}
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:
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)
, +(s(x), s(y)) -> s(+(s(x), +(y, 0())))
, +^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)}
Details:
Interpretation Functions:
+(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
+^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{+^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
and weakly orienting the rules
{ +(0(), y) -> y
, +(s(x), 0()) -> s(x)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{+^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))}
Details:
Interpretation Functions:
+(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [1]
s(x1) = [1] x1 + [4]
+^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_0() = [0]
c_1() = [0]
c_2(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: {+(s(x), s(y)) -> s(+(s(x), +(y, 0())))}
Weak Rules:
{ +^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))
, +(0(), y) -> y
, +(s(x), 0()) -> s(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: {+(s(x), s(y)) -> s(+(s(x), +(y, 0())))}
Weak Rules:
{ +^#(s(x), s(y)) -> c_2(+^#(s(x), +(y, 0())))
, +(0(), y) -> y
, +(s(x), 0()) -> s(x)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ +_0(2, 2) -> 6
, +_0(3, 2) -> 6
, 0_0() -> 2
, 0_0() -> 6
, s_0(2) -> 3
, s_0(2) -> 6
, s_0(3) -> 3
, s_0(3) -> 6
, +^#_0(2, 2) -> 4
, +^#_0(2, 3) -> 4
, +^#_0(3, 2) -> 4
, +^#_0(3, 3) -> 4
, +^#_0(3, 6) -> 5
, c_2_0(5) -> 4
, c_2_0(5) -> 5}
3) {+^#(0(), y) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(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: {+^#(0(), y) -> c_0()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{+^#(0(), y) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{+^#(0(), y) -> c_0()}
Details:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
c_2(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: {+^#(0(), y) -> c_0()}
Details:
The given problem does not contain any strict rules