'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, a__b^#() -> c_1()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark^#(b()) -> c_3(a__b^#())
, mark^#(a()) -> c_4()
, a__f^#(X1, X2, X3) -> c_5()
, a__b^#() -> c_6()}
The usable rules are:
{ a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
The estimated dependency graph contains the following edges:
{a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
==> {a__f^#(X1, X2, X3) -> c_5()}
{a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
==> {a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
==> {a__f^#(X1, X2, X3) -> c_5()}
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
==> {a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
{mark^#(b()) -> c_3(a__b^#())}
==> {a__b^#() -> c_6()}
{mark^#(b()) -> c_3(a__b^#())}
==> {a__b^#() -> c_1()}
We consider the following path(s):
1) { mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
The usable rules for this path are the following:
{ a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
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:
{ a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(b()) -> a__b()
, mark(a()) -> a()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [1] x1 + [1]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
and weakly orienting the rules
{ mark(b()) -> a__b()
, mark(a()) -> a()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [1]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__b() -> a()}
and weakly orienting the rules
{ mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__b() -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [8]
b() = [8]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [1] x1 + [4]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__b() -> b()}
and weakly orienting the rules
{ a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__b() -> b()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [7]
b() = [6]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4]
c_0(x1) = [1] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
and weakly orienting the rules
{ a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [1]
a__b() = [1]
b() = [1]
mark(x1) = [1] x1 + [0]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [15]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
a) We first check the conditional [Fail]:
We are not considering a strict trs contains single rule TRS.
b) We continue with the else-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 relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [4] x1 + [1] x2 + [4] x3 + [4]
a() = [1]
a__b() = [1]
b() = [0]
mark(x1) = [4] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4]
a__f^#(x1, x2, x3) = [2] x1 + [1] x2 + [4] x3 + [7]
c_0(x1) = [1] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [7] x1 + [5]
c_2(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(X1, X2, X3) -> f(X1, X2, X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(X1, X2, X3) -> f(X1, X2, X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
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 relative runtime-complexity with respect to
Strict Rules: {a__f(X1, X2, X3) -> f(X1, X2, X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(X1, X2, X3) -> f(X1, X2, X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [4] x1 + [1] x2 + [4] x3 + [6]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [4] x1 + [3]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [5]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [4] x3 + [1]
c_0(x1) = [1] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [7] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
2) { mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, a__f^#(X1, X2, X3) -> c_5()}
The usable rules for this path are the following:
{ a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
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:
{ a__b() -> a()
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [7]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
and weakly orienting the rules
{ mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__b() -> a()}
and weakly orienting the rules
{ mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__b() -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [8]
b() = [8]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [3]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [3]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__b() -> b()}
and weakly orienting the rules
{ a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__b() -> b()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [11]
b() = [10]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0(x1) = [1] x1 + [1]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
and weakly orienting the rules
{ a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [0]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [1] x1 + [5]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
a) We first check the conditional [Fail]:
We are not considering a strict trs contains single rule TRS.
b) We continue with the else-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 relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [4] x1 + [1] x2 + [4] x3 + [4]
a() = [5]
a__b() = [5]
b() = [0]
mark(x1) = [4] x1 + [7]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
a__f^#(x1, x2, x3) = [4] x1 + [1] x2 + [7] x3 + [5]
c_0(x1) = [2] x1 + [1]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [7] x1 + [5]
c_2(x1) = [1] x1 + [5]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(X1, X2, X3) -> f(X1, X2, X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(X1, X2, X3) -> f(X1, X2, X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
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 relative runtime-complexity with respect to
Strict Rules: {a__f(X1, X2, X3) -> f(X1, X2, X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(X1, X2, X3) -> f(X1, X2, X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f^#(a(), X, X) -> c_0(a__f^#(X, a__b(), b()))
, mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__f^#(X1, X2, X3) -> c_5()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [4] x1 + [1] x2 + [4] x3 + [2]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [4] x1 + [2]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [7] x3 + [0]
c_0(x1) = [1] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [7] x1 + [4]
c_2(x1) = [1] x1 + [5]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
3) {mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
The usable rules for this path are the following:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
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:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(b()) -> a__b()
, mark(a()) -> a()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
and weakly orienting the rules
{ mark(b()) -> a__b()
, mark(a()) -> a()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [3]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__b() -> a()}
and weakly orienting the rules
{ mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__b() -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [8]
b() = [8]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__b() -> b()}
and weakly orienting the rules
{ a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__b() -> b()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [1]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(X1, X2, X3) -> f(X1, X2, X3)}
and weakly orienting the rules
{ a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(X1, X2, X3) -> f(X1, X2, X3)}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
a() = [0]
a__b() = [8]
b() = [8]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
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 relative runtime-complexity with respect to
Strict Rules: {mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [4] x1 + [1] x2 + [4] x3 + [3]
a() = [4]
a__b() = [4]
b() = [2]
mark(x1) = [4] x1 + [0]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [3]
a__f^#(x1, x2, x3) = [4] x1 + [1] x2 + [2] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [6] x1 + [3]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
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 relative runtime-complexity with respect to
Strict Rules: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, a__b() -> b()
, a__b() -> a()
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, mark(b()) -> a__b()
, mark(a()) -> a()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [6] x1 + [1] x2 + [5] x3 + [0]
a() = [2]
a__b() = [4]
b() = [0]
mark(x1) = [6] x1 + [4]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [2] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [6] x1 + [6]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
4) { mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()}
The usable rules for this path are the following:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)}
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:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()
, a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [4]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(b()) -> a__b()}
and weakly orienting the rules
{ mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(b()) -> a__b()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f^#(X1, X2, X3) -> c_5()}
and weakly orienting the rules
{ mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(X1, X2, X3) -> c_5()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
and weakly orienting the rules
{ a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(X1, X2, X3) -> f(X1, X2, X3)}
and weakly orienting the rules
{ mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(X1, X2, X3) -> f(X1, X2, X3)}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [8]
a() = [0]
a__b() = [8]
b() = [8]
mark(x1) = [1] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
a__f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
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 relative runtime-complexity with respect to
Strict Rules: {mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)}
Weak Rules:
{ a__f(a(), X, X) -> a__f(X, a__b(), b())
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [2] x1 + [1] x2 + [4] x3 + [2]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [4] x1 + [1]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
a__f^#(x1, x2, x3) = [4] x1 + [1] x2 + [4] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [5] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
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 relative runtime-complexity with respect to
Strict Rules: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {a__f(a(), X, X) -> a__f(X, a__b(), b())}
Weak Rules:
{ mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
, a__f(X1, X2, X3) -> f(X1, X2, X3)
, mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
, a__f^#(X1, X2, X3) -> c_5()
, mark(b()) -> a__b()
, mark(a()) -> a()
, a__b() -> a()
, a__b() -> b()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [4] x1 + [1] x2 + [4] x3 + [4]
a() = [1]
a__b() = [1]
b() = [0]
mark(x1) = [4] x1 + [5]
f(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [3]
a__f^#(x1, x2, x3) = [2] x1 + [1] x2 + [4] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [4] x1 + [5]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [0]
5) { mark^#(b()) -> c_3(a__b^#())
, a__b^#() -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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: {a__b^#() -> c_1()}
Weak Rules: {mark^#(b()) -> c_3(a__b^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__b^#() -> c_1()}
and weakly orienting the rules
{mark^#(b()) -> c_3(a__b^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__b^#() -> c_1()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [1]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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:
{ a__b^#() -> c_1()
, mark^#(b()) -> c_3(a__b^#())}
Details:
The given problem does not contain any strict rules
6) { mark^#(b()) -> c_3(a__b^#())
, a__b^#() -> c_6()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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: {a__b^#() -> c_6()}
Weak Rules: {mark^#(b()) -> c_3(a__b^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__b^#() -> c_6()}
and weakly orienting the rules
{mark^#(b()) -> c_3(a__b^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__b^#() -> c_6()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [1]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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:
{ a__b^#() -> c_6()
, mark^#(b()) -> c_3(a__b^#())}
Details:
The given problem does not contain any strict rules
7) {mark^#(b()) -> c_3(a__b^#())}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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: {mark^#(b()) -> c_3(a__b^#())}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(b()) -> c_3(a__b^#())}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(b()) -> c_3(a__b^#())}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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: {mark^#(b()) -> c_3(a__b^#())}
Details:
The given problem does not contain any strict rules
8) {mark^#(a()) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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: {mark^#(a()) -> c_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(a()) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(a()) -> c_4()}
Details:
Interpretation Functions:
a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
a__b() = [0]
b() = [0]
mark(x1) = [0] x1 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
a__b^#() = [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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: {mark^#(a()) -> c_4()}
Details:
The given problem does not contain any strict rules