'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(c(X, s(Y))) -> f(c(s(X), Y))
, g(c(s(X), Y)) -> f(c(X, s(Y)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))
, g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
==> {f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
{g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
==> {f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
We consider the following path(s):
1) { g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))
, f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), 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]
c(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
g(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]
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^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
Weak Rules: {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
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: {f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
Weak Rules: {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
Weak Rules: {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
Weak Rules: {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
Weak Rules: {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Details:
a) We first check the conditional [Success]:
We are considering a strict trs contains single rule TRS.
b) We continue with the then-branch:
The problem was solved by processor 'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation'':
'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation''
--------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
Weak Rules: {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(X, s(Y))) -> c_0(f^#(c(s(X), Y)))}
Weak Rules: {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [1] x2 + [6]
s(x1) = [1] x1 + [3]
g(x1) = [0] x1 + [0]
f^#(x1) = [4] x1 + [1]
c_0(x1) = [1] x1 + [7]
g^#(x1) = [7] x1 + [1]
c_1(x1) = [1] x1 + [0]
2) {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(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]
c(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
g(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]
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^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_1(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: {g^#(c(s(X), Y)) -> c_1(f^#(c(X, s(Y))))}
Details:
The given problem does not contain any strict rules