'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(X) -> f(X)
, a__p(X) -> p(X)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a__f^#(0()) -> c_0()
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))
, a__p^#(s(0())) -> c_2()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark^#(0()) -> c_5()
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(X) -> c_8()
, a__p^#(X) -> c_9()}
The usable rules are:
{ a__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
The estimated dependency graph contains the following edges:
{a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
==> {a__f^#(X) -> c_8()}
{a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
==> {a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
{a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
==> {a__f^#(0()) -> c_0()}
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
==> {a__f^#(X) -> c_8()}
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
==> {a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
==> {a__f^#(0()) -> c_0()}
{mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
==> {a__p^#(X) -> c_9()}
{mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
==> {a__p^#(s(0())) -> c_2()}
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
==> {mark^#(s(X)) -> c_7(mark^#(X))}
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
==> {mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
==> {mark^#(0()) -> c_5()}
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
==> {mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
==> {mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
{mark^#(s(X)) -> c_7(mark^#(X))}
==> {mark^#(s(X)) -> c_7(mark^#(X))}
{mark^#(s(X)) -> c_7(mark^#(X))}
==> {mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
{mark^#(s(X)) -> c_7(mark^#(X))}
==> {mark^#(0()) -> c_5()}
{mark^#(s(X)) -> c_7(mark^#(X))}
==> {mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
{mark^#(s(X)) -> c_7(mark^#(X))}
==> {mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
We consider the following path(s):
1) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(X) -> c_8()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
and weakly orienting the rules
{ mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [4]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(p(X)) -> a__p(mark(X))}
and weakly orienting the rules
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(p(X)) -> a__p(mark(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(f(X)) -> a__f(mark(X))}
and weakly orienting the rules
{ mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(f(X)) -> a__f(mark(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [12]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
f(x1) = [1] x1 + [15]
s(x1) = [1] x1 + [1]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [8]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
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^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
Details:
Interpretation Functions:
a__f(x1) = [1 5] x1 + [0]
[0 0] [2]
0() = [0]
[6]
cons(x1, x2) = [1 0] x1 + [1 5] x2 + [1]
[0 0] [0 0] [0]
f(x1) = [1 1] x1 + [0]
[0 0] [2]
s(x1) = [1 1] x1 + [0]
[0 1] [6]
a__p(x1) = [1 0] x1 + [2]
[0 0] [7]
mark(x1) = [5 0] x1 + [0]
[0 1] [0]
p(x1) = [1 0] x1 + [1]
[0 0] [7]
a__f^#(x1) = [1 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
a__p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
mark^#(x1) = [5 0] x1 + [0]
[2 6] [4]
c_3(x1) = [1 0] x1 + [0]
[0 0] [1]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [1]
[0 0] [1]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
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(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
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:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
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(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(X) -> c_8()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [4]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [2]
s(x1) = [1] x1 + [1]
a__p(x1) = [1] x1 + [0]
mark(x1) = [4] x1 + [4]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [4] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
2) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(X) -> c_9()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [1]
c_2() = [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
and weakly orienting the rules
{ mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [4]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(p(X)) -> a__p(mark(X))}
and weakly orienting the rules
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(p(X)) -> a__p(mark(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [8]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [6]
c_2() = [0]
mark^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(f(X)) -> a__f(mark(X))}
and weakly orienting the rules
{ mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(f(X)) -> a__f(mark(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [12]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [9]
c_2() = [0]
mark^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [3]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
f(x1) = [1] x1 + [15]
s(x1) = [1] x1 + [1]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [8]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [2]
c_2() = [0]
mark^#(x1) = [1] x1 + [15]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
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^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
Details:
Interpretation Functions:
a__f(x1) = [1 4] x1 + [1]
[0 1] [5]
0() = [0]
[2]
cons(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
f(x1) = [1 1] x1 + [1]
[0 1] [5]
s(x1) = [1 0] x1 + [0]
[0 1] [4]
a__p(x1) = [1 1] x1 + [1]
[0 0] [4]
mark(x1) = [4 1] x1 + [0]
[0 1] [2]
p(x1) = [1 1] x1 + [0]
[0 0] [3]
a__f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
a__p^#(x1) = [1 0] x1 + [3]
[0 0] [0]
c_2() = [0]
[0]
mark^#(x1) = [4 2] x1 + [1]
[0 0] [2]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [2]
[0 0] [1]
c_5() = [0]
[0]
c_6(x1) = [1 4] x1 + [0]
[0 0] [1]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
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(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
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:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
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(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, mark(0()) -> 0()
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(X) -> c_9()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [4]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [2]
s(x1) = [1] x1 + [1]
a__p(x1) = [1] x1 + [0]
mark(x1) = [4] x1 + [0]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [2]
c_2() = [0]
mark^#(x1) = [4] x1 + [3]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
3) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [3]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [6]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [8]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [6]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(f(X)) -> a__f(mark(X))}
and weakly orienting the rules
{ mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(f(X)) -> a__f(mark(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [2]
a__f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
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^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1 6] x1 + [1]
[0 0] [1]
0() = [0]
[6]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [1]
f(x1) = [1 3] x1 + [1]
[0 0] [1]
s(x1) = [1 0] x1 + [0]
[0 1] [5]
a__p(x1) = [1 0] x1 + [4]
[0 0] [6]
mark(x1) = [2 0] x1 + [0]
[0 1] [0]
p(x1) = [1 0] x1 + [2]
[0 0] [6]
a__f^#(x1) = [1 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
a__p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
mark^#(x1) = [2 0] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
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(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
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:
{ mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
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(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [5]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [2]
s(x1) = [1] x1 + [1]
a__p(x1) = [1] x1 + [0]
mark(x1) = [4] x1 + [4]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [4] x1 + [1]
c_3(x1) = [1] x1 + [2]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
4) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p^#(s(0())) -> c_2()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [10]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__p(X) -> p(X)}
and weakly orienting the rules
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__p(X) -> p(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [14]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [10]
a__p(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [7]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
and weakly orienting the rules
{ a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [3]
c_2() = [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [11]
c_7(x1) = [1] x1 + [8]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark(f(X)) -> a__f(mark(X))}
and weakly orienting the rules
{ mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(f(X)) -> a__f(mark(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [2]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [6]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [8]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [2]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
f(x1) = [1] x1 + [14]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [15]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
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^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
Details:
Interpretation Functions:
a__f(x1) = [1 0 1] x1 + [0]
[0 1 1] [1]
[0 0 0] [1]
0() = [0]
[0]
[0]
cons(x1, x2) = [1 0 1] x1 + [1 0 0] x2 + [0]
[0 1 1] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
f(x1) = [1 0 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [1]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [1]
[0 0 0] [1]
a__p(x1) = [1 0 0] x1 + [0]
[0 1 0] [1]
[0 0 0] [0]
mark(x1) = [1 1 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
p(x1) = [1 0 0] x1 + [0]
[0 1 0] [1]
[0 0 0] [0]
a__f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a__p^#(x1) = [1 0 0] x1 + [0]
[0 0 1] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
mark^#(x1) = [1 1 0] x1 + [1]
[1 1 0] [1]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
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:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(f(X)) -> a__f(mark(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, a__p(X) -> p(X)
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p^#(s(0())) -> c_2()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [4]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [1]
a__p(x1) = [1] x1 + [0]
mark(x1) = [4] x1 + [4]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [4] x1 + [5]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
5) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [3]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
and weakly orienting the rules
{ mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(p(X)) -> c_4(a__p^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
f(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [6]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [4]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [6]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
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^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1 1 0] x1 + [1]
[0 0 0] [0]
[0 1 1] [1]
0() = [0]
[0]
[0]
cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
f(x1) = [1 0 0] x1 + [1]
[0 0 0] [0]
[0 0 1] [1]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 0 1] [1]
a__p(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [1]
mark(x1) = [1 0 1] x1 + [0]
[0 0 0] [1]
[1 0 1] [0]
p(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [1]
a__f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a__p^#(x1) = [1 0 0] x1 + [0]
[0 1 0] [1]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
mark^#(x1) = [1 0 1] x1 + [0]
[0 0 0] [1]
[0 0 1] [1]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 1 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
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:
{ mark(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Weak Rules:
{ mark(p(X)) -> a__p(mark(X))
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, mark(f(X)) -> a__f(mark(X))
, a__p(X) -> p(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(p(X)) -> c_4(a__p^#(mark(X)))
, mark(0()) -> 0()
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [7]
0() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
f(x1) = [1] x1 + [2]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [4] x1 + [0]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [1] x1 + [0]
c_2() = [0]
mark^#(x1) = [4] x1 + [5]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [2]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
6) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))
, a__f^#(0()) -> c_0()}
The usable rules for this path are the following:
{ a__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
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__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(0()) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [1]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f^#(0()) -> c_0()}
and weakly orienting the rules
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(0()) -> c_0()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [2]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ a__f^#(0()) -> c_0()
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [7]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [8]
c_7(x1) = [1] x1 + [1]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(0()) -> c_0()
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [1]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [4]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [8]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(s(X)) -> c_7(mark^#(X))}
and weakly orienting the rules
{ a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(0()) -> c_0()
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [7]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [11]
a__p(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [3]
a__f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [5]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(0()) -> cons(0(), f(s(0())))}
and weakly orienting the rules
{ mark^#(s(X)) -> c_7(mark^#(X))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(0()) -> c_0()
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(0()) -> cons(0(), f(s(0())))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [4]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [12]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [10]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [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(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
Weak Rules:
{ a__f(0()) -> cons(0(), f(s(0())))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(0()) -> c_0()
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(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:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
Weak Rules:
{ a__f(0()) -> cons(0(), f(s(0())))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f(X) -> f(X)
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(0()) -> c_0()
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, a__f_2(11) -> 4
, a__f_2(11) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, 0_1() -> 7
, 0_1() -> 9
, 0_2() -> 11
, 0_2() -> 13
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, cons_2(13, 15) -> 4
, cons_2(13, 15) -> 5
, f_0(2) -> 2
, f_1(5) -> 4
, f_1(5) -> 5
, f_2(11) -> 4
, f_2(11) -> 5
, f_2(12) -> 15
, s_0(2) -> 2
, s_1(5) -> 4
, s_1(5) -> 5
, s_1(9) -> 8
, s_2(13) -> 12
, a__p_1(5) -> 4
, a__p_1(5) -> 5
, a__p_1(8) -> 7
, a__p_2(12) -> 11
, mark_0(2) -> 4
, mark_1(2) -> 5
, p_0(2) -> 2
, p_1(5) -> 4
, p_1(5) -> 5
, p_1(8) -> 7
, p_2(12) -> 11
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 10
, a__f^#_1(7) -> 6
, a__f^#_2(11) -> 14
, c_0_0() -> 1
, c_0_0() -> 3
, c_0_1() -> 3
, c_0_1() -> 6
, c_0_1() -> 10
, c_0_2() -> 14
, c_1_1(6) -> 1
, c_1_1(10) -> 3
, c_1_2(14) -> 10
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(10) -> 1
, c_6_0(1) -> 1
, c_7_0(1) -> 1}
7) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))
, a__f^#(X) -> c_8()}
The usable rules for this path are the following:
{ a__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
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__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(X) -> c_8()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)
, a__f^#(X) -> c_8()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)
, a__f^#(X) -> c_8()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [1]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)
, a__f^#(X) -> c_8()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [6]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [8]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(X) -> f(X)
, mark^#(s(X)) -> c_7(mark^#(X))}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)
, a__f^#(X) -> c_8()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(X) -> f(X)
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [8]
0() = [11]
cons(x1, x2) = [1] x1 + [1] x2 + [6]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [6]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [12]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(0()) -> cons(0(), f(s(0())))}
and weakly orienting the rules
{ a__f(X) -> f(X)
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)
, a__f^#(X) -> c_8()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(0()) -> cons(0(), f(s(0())))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [8]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [4]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [2]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [9]
c_7(x1) = [1] x1 + [2]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)
, a__f^#(X) -> c_8()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [4]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [6]
c_3(x1) = [1] x1 + [5]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [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(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)
, a__f^#(X) -> c_8()}
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:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)
, a__f^#(X) -> c_8()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, a__f_2(11) -> 4
, a__f_2(11) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, 0_1() -> 7
, 0_1() -> 9
, 0_2() -> 11
, 0_2() -> 13
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, cons_2(13, 15) -> 4
, cons_2(13, 15) -> 5
, f_0(2) -> 2
, f_1(5) -> 4
, f_1(5) -> 5
, f_2(11) -> 4
, f_2(11) -> 5
, f_2(12) -> 15
, s_0(2) -> 2
, s_1(5) -> 4
, s_1(5) -> 5
, s_1(9) -> 8
, s_2(13) -> 12
, a__p_1(5) -> 4
, a__p_1(5) -> 5
, a__p_1(8) -> 7
, a__p_2(12) -> 11
, mark_0(2) -> 4
, mark_1(2) -> 5
, p_0(2) -> 2
, p_1(5) -> 4
, p_1(5) -> 5
, p_1(8) -> 7
, p_2(12) -> 11
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 10
, a__f^#_1(7) -> 6
, a__f^#_2(11) -> 14
, c_1_1(6) -> 1
, c_1_1(10) -> 3
, c_1_2(14) -> 10
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(10) -> 1
, c_6_0(1) -> 1
, c_7_0(1) -> 1
, c_8_0() -> 1
, c_8_0() -> 3
, c_8_1() -> 6
, c_8_1() -> 10
, c_8_2() -> 14}
8) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
The usable rules for this path are the following:
{ a__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
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__p(s(0())) -> 0()
, mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [1]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(s(X)) -> c_7(mark^#(X))}
and weakly orienting the rules
{ a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{ mark^#(s(X)) -> c_7(mark^#(X))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [2]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [15]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__f(X) -> f(X)}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [7]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [3]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__f(0()) -> cons(0(), f(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__f(0()) -> cons(0(), f(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [15]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [12]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [2]
a__p(x1) = [1] x1 + [4]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [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(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
Weak Rules:
{ a__f(0()) -> cons(0(), f(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(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:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f^#(s(0())) -> c_1(a__f^#(a__p(s(0()))))}
Weak Rules:
{ a__f(0()) -> cons(0(), f(s(0())))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__p(s(0())) -> 0()
, mark(0()) -> 0()
, a__p(X) -> p(X)}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, a__f_2(11) -> 4
, a__f_2(11) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, 0_1() -> 7
, 0_1() -> 9
, 0_2() -> 11
, 0_2() -> 13
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, cons_2(13, 15) -> 4
, cons_2(13, 15) -> 5
, f_0(2) -> 2
, f_1(5) -> 4
, f_1(5) -> 5
, f_2(11) -> 4
, f_2(11) -> 5
, f_2(12) -> 15
, s_0(2) -> 2
, s_1(5) -> 4
, s_1(5) -> 5
, s_1(9) -> 8
, s_2(13) -> 12
, a__p_1(5) -> 4
, a__p_1(5) -> 5
, a__p_1(8) -> 7
, a__p_2(12) -> 11
, mark_0(2) -> 4
, mark_1(2) -> 5
, p_0(2) -> 2
, p_1(5) -> 4
, p_1(5) -> 5
, p_1(8) -> 7
, p_2(12) -> 11
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 10
, a__f^#_1(7) -> 6
, a__f^#_2(11) -> 14
, c_1_1(6) -> 1
, c_1_1(10) -> 3
, c_1_2(14) -> 10
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(10) -> 1
, c_6_0(1) -> 1
, c_7_0(1) -> 1}
9) { mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, a__f^#(0()) -> c_0()}
The usable rules for this path are the following:
{ mark(f(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)}
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(X)) -> a__f(mark(X))
, mark(p(X)) -> a__p(mark(X))
, mark(0()) -> 0()
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(s(0())) -> 0()
, a__p(X) -> p(X)
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(s(0())) -> a__f(a__p(s(0())))
, a__f(X) -> f(X)
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(0()) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark(0()) -> 0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark(0()) -> 0()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
and weakly orienting the rules
{mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(mark(X)))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [2]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [8]
c_7(x1) = [1] x1 + [10]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(0()) -> c_0()}
and weakly orienting the rules
{ mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(0()) -> c_0()}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [0]
0() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [9]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [8]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ mark(p(X)) -> a__p(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
and weakly orienting the rules
{ a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(0()) -> c_0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(p(X)) -> a__p(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [1]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [5]
mark(x1) = [1] x1 + [1]
p(x1) = [1] x1 + [10]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [3]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{ mark(p(X)) -> a__p(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(0()) -> c_0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [1] x1 + [8]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [8]
a__f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [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(X)) -> a__f(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(p(X)) -> a__p(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(0()) -> c_0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()}
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:
{ mark(f(X)) -> a__f(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__p(X) -> p(X)
, a__f(s(0())) -> a__f(a__p(s(0())))}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark(p(X)) -> a__p(mark(X))
, a__f(0()) -> cons(0(), f(s(0())))
, a__f(X) -> f(X)
, a__p(s(0())) -> 0()
, mark^#(s(X)) -> c_7(mark^#(X))
, a__f^#(0()) -> c_0()
, mark^#(f(X)) -> c_3(a__f^#(mark(X)))
, mark(0()) -> 0()}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ a__f_1(5) -> 4
, a__f_1(5) -> 5
, a__f_2(7) -> 4
, a__f_2(7) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, 0_2() -> 7
, 0_2() -> 9
, 0_3() -> 11
, 0_3() -> 14
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, cons_2(9, 10) -> 4
, cons_2(9, 10) -> 5
, cons_3(11, 12) -> 4
, cons_3(14, 12) -> 5
, f_0(2) -> 2
, f_1(5) -> 4
, f_1(5) -> 5
, f_2(7) -> 4
, f_2(7) -> 5
, f_2(8) -> 10
, f_3(13) -> 12
, s_0(2) -> 2
, s_1(5) -> 4
, s_1(5) -> 5
, s_2(9) -> 8
, s_3(14) -> 13
, a__p_0(4) -> 4
, a__p_1(5) -> 5
, a__p_2(8) -> 7
, mark_0(2) -> 4
, mark_1(2) -> 5
, p_0(2) -> 2
, p_1(4) -> 4
, p_2(5) -> 5
, p_3(8) -> 7
, a__f^#_0(2) -> 1
, a__f^#_0(4) -> 3
, a__f^#_1(5) -> 6
, c_0_0() -> 1
, c_0_0() -> 3
, c_0_1() -> 3
, c_0_1() -> 6
, mark^#_0(2) -> 1
, c_3_0(3) -> 1
, c_3_1(6) -> 1
, c_6_0(1) -> 1
, c_7_0(1) -> 1}
10)
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))}
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) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
a__p(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [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^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(s(X)) -> c_7(mark^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(s(X)) -> c_7(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [1]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
and weakly orienting the rules
{mark^#(s(X)) -> c_7(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(cons(X1, X2)) -> c_6(mark^#(X1))}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [8]
a__p(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [3]
c_7(x1) = [1] x1 + [1]
c_8() = [0]
c_9() = [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^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
The given problem does not contain any strict rules
11)
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))
, mark^#(0()) -> c_5()}
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) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
a__p(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [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^#(0()) -> c_5()}
Weak Rules:
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(0()) -> c_5()}
and weakly orienting the rules
{ mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(0()) -> c_5()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
a__p(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
a__p^#(x1) = [0] x1 + [0]
c_2() = [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [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^#(0()) -> c_5()
, mark^#(cons(X1, X2)) -> c_6(mark^#(X1))
, mark^#(s(X)) -> c_7(mark^#(X))}
Details:
The given problem does not contain any strict rules