'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(s(X)) -> f(X)
, g(cons(0(), Y)) -> g(Y)
, g(cons(s(X), Y)) -> s(X)
, h(cons(X, Y)) -> h(g(cons(X, Y)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(s(X)) -> c_0(f^#(X))
, g^#(cons(0(), Y)) -> c_1(g^#(Y))
, g^#(cons(s(X), Y)) -> c_2()
, h^#(cons(X, Y)) -> c_3(h^#(g(cons(X, Y))))}
The usable rules are:
{ g(cons(0(), Y)) -> g(Y)
, g(cons(s(X), Y)) -> s(X)}
The estimated dependency graph contains the following edges:
{f^#(s(X)) -> c_0(f^#(X))}
==> {f^#(s(X)) -> c_0(f^#(X))}
{g^#(cons(0(), Y)) -> c_1(g^#(Y))}
==> {g^#(cons(s(X), Y)) -> c_2()}
{g^#(cons(0(), Y)) -> c_1(g^#(Y))}
==> {g^#(cons(0(), Y)) -> c_1(g^#(Y))}
We consider the following path(s):
1) {h^#(cons(X, Y)) -> c_3(h^#(g(cons(X, Y))))}
The usable rules for this path are the following:
{ g(cons(0(), Y)) -> g(Y)
, g(cons(s(X), Y)) -> s(X)}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [4]
g(x1) = [1] x1 + [9]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [8]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
h^#(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: {h^#(cons(X, Y)) -> c_3(h^#(g(cons(X, Y))))}
Weak Rules:
{ g(cons(0(), Y)) -> g(Y)
, g(cons(s(X), Y)) -> s(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: {h^#(cons(X, Y)) -> c_3(h^#(g(cons(X, Y))))}
Weak Rules:
{ g(cons(0(), Y)) -> g(Y)
, g(cons(s(X), Y)) -> 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 DP runtime-complexity with respect to
Strict Rules: {h^#(cons(X, Y)) -> c_3(h^#(g(cons(X, Y))))}
Weak Rules:
{ g(cons(0(), Y)) -> g(Y)
, g(cons(s(X), Y)) -> s(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ s_0(2) -> 2
, s_1(2) -> 4
, g_1(2) -> 4
, g_1(5) -> 4
, cons_0(2, 2) -> 2
, cons_1(2, 2) -> 5
, 0_0() -> 2
, h^#_0(2) -> 1
, h^#_1(4) -> 3
, c_3_1(3) -> 1}
2) { g^#(cons(0(), Y)) -> c_1(g^#(Y))
, g^#(cons(s(X), Y)) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
h^#(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: {g^#(cons(s(X), Y)) -> c_2()}
Weak Rules: {g^#(cons(0(), Y)) -> c_1(g^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(cons(s(X), Y)) -> c_2()}
and weakly orienting the rules
{g^#(cons(0(), Y)) -> c_1(g^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(cons(s(X), Y)) -> c_2()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
h^#(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:
{ g^#(cons(s(X), Y)) -> c_2()
, g^#(cons(0(), Y)) -> c_1(g^#(Y))}
Details:
The given problem does not contain any strict rules
3) {f^#(s(X)) -> c_0(f^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
h^#(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: {f^#(s(X)) -> c_0(f^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(s(X)) -> c_0(f^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(X)) -> c_0(f^#(X))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [8]
g(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [3]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
h^#(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: {f^#(s(X)) -> c_0(f^#(X))}
Details:
The given problem does not contain any strict rules
4) {g^#(cons(0(), Y)) -> c_1(g^#(Y))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
h^#(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: {g^#(cons(0(), Y)) -> c_1(g^#(Y))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(cons(0(), Y)) -> c_1(g^#(Y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(cons(0(), Y)) -> c_1(g^#(Y))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
h(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [3]
c_2() = [0]
h^#(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: {g^#(cons(0(), Y)) -> c_1(g^#(Y))}
Details:
The given problem does not contain any strict rules