'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ -^#(0(), y) -> c_0()
, -^#(x, 0()) -> c_1()
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, p^#(0()) -> c_3()
, p^#(s(x)) -> c_4()}
The usable rules are:
{ p(0()) -> 0()
, p(s(x)) -> x}
The estimated dependency graph contains the following edges:
{-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
==> {-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
{-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
==> {-^#(x, 0()) -> c_1()}
{-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
==> {-^#(0(), y) -> c_0()}
We consider the following path(s):
1) { -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, -^#(0(), y) -> c_0()}
The usable rules for this path are the following:
{ p(0()) -> 0()
, p(s(x)) -> x}
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) = [1] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [1] x1 + [1]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [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:
{ p(0()) -> 0()
, p(s(x)) -> x
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{-^#(0(), y) -> c_0()}
and weakly orienting the rules
{ p(0()) -> 0()
, p(s(x)) -> x
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
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() = [1]
s(x1) = [1] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [1] x1 + [0]
-^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [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()
, p(0()) -> 0()
, p(s(x)) -> x
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
Details:
The given problem does not contain any strict rules
2) { -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, -^#(x, 0()) -> c_1()}
The usable rules for this path are the following:
{ p(0()) -> 0()
, p(s(x)) -> x}
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) = [1] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [1] x1 + [1]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [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: {-^#(x, 0()) -> c_1()}
Weak Rules:
{ p(0()) -> 0()
, p(s(x)) -> x
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{-^#(x, 0()) -> c_1()}
and weakly orienting the rules
{ p(0()) -> 0()
, p(s(x)) -> x
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{-^#(x, 0()) -> c_1()}
Details:
Interpretation Functions:
-(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [1]
s(x1) = [1] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [1] x1 + [0]
-^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [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:
{ -^#(x, 0()) -> c_1()
, p(0()) -> 0()
, p(s(x)) -> x
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
Details:
The given problem does not contain any strict rules
3) {-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
The usable rules for this path are the following:
{ p(0()) -> 0()
, p(s(x)) -> x}
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) = [1] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [1] x1 + [1]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [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: {-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
Weak Rules:
{ p(0()) -> 0()
, p(s(x)) -> x}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
Weak Rules:
{ p(0()) -> 0()
, p(s(x)) -> x}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
Weak Rules:
{ p(0()) -> 0()
, p(s(x)) -> x}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 0_0() -> 2
, 0_0() -> 9
, s_0(2) -> 3
, s_0(2) -> 9
, s_0(3) -> 3
, s_0(3) -> 9
, s_1(2) -> 10
, s_1(3) -> 10
, p_1(10) -> 9
, -^#_0(2, 2) -> 7
, -^#_0(2, 3) -> 7
, -^#_0(3, 2) -> 7
, -^#_0(3, 3) -> 7
, -^#_1(2, 9) -> 8
, -^#_1(3, 9) -> 8
, c_2_1(8) -> 7
, c_2_1(8) -> 8}
4) {p^#(0()) -> c_3()}
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]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [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: {p^#(0()) -> c_3()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{p^#(0()) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{p^#(0()) -> c_3()}
Details:
Interpretation Functions:
-(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
p^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4() = [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: {p^#(0()) -> c_3()}
Details:
The given problem does not contain any strict rules
5) {p^#(s(x)) -> c_4()}
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]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [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: {p^#(s(x)) -> c_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{p^#(s(x)) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{p^#(s(x)) -> c_4()}
Details:
Interpretation Functions:
-(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
greater(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
-^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
p^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4() = [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: {p^#(s(x)) -> c_4()}
Details:
The given problem does not contain any strict rules