'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, mark(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> zeros()
, a__tail(X) -> tail(X)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a__zeros^#() -> c_0()
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(zeros()) -> c_2(a__zeros^#())
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(0()) -> c_5()
, a__zeros^#() -> c_6()
, a__tail^#(X) -> c_7()}
The usable rules are:
{ mark(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(X)}
The estimated dependency graph contains the following edges:
{a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))}
==> {mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
{a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))}
==> {mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
{a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))}
==> {mark^#(zeros()) -> c_2(a__zeros^#())}
{a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))}
==> {mark^#(0()) -> c_5()}
{mark^#(zeros()) -> c_2(a__zeros^#())}
==> {a__zeros^#() -> c_6()}
{mark^#(zeros()) -> c_2(a__zeros^#())}
==> {a__zeros^#() -> c_0()}
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
==> {a__tail^#(X) -> c_7()}
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
==> {a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))}
{mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
==> {mark^#(0()) -> c_5()}
{mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
==> {mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
{mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
==> {mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
{mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
==> {mark^#(zeros()) -> c_2(a__zeros^#())}
We consider the following path(s):
1) { a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros^#() -> c_6()}
The usable rules for this path are the following:
{ mark(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(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(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(X)
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__zeros^#() -> c_6()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__zeros() -> zeros()}
and weakly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__zeros() -> zeros()}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(zeros()) -> c_2(a__zeros^#())}
and weakly orienting the rules
{ a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(zeros()) -> c_2(a__zeros^#())}
Details:
Interpretation Functions:
a__zeros() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [8]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [3]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__zeros^#() -> c_6()}
and weakly orienting the rules
{ mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__zeros^#() -> c_6()}
Details:
Interpretation Functions:
a__zeros() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [8]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [8]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [3]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [7]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
and weakly orienting the rules
{ a__zeros^#() -> c_6()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
Details:
Interpretation Functions:
a__zeros() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [8]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [8]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__tail(X) -> tail(X)}
and weakly orienting the rules
{ mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros^#() -> c_6()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__tail(X) -> tail(X)}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [0]
zeros() = [1]
a__tail(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__zeros() -> cons(0(), zeros())}
and weakly orienting the rules
{ a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros^#() -> c_6()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__zeros() -> cons(0(), zeros())}
Details:
Interpretation Functions:
a__zeros() = [4]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
zeros() = [2]
a__tail(x1) = [1] x1 + [7]
mark(x1) = [1] x1 + [2]
tail(x1) = [1] x1 + [5]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [5]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [5]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [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(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)}
Weak Rules:
{ a__zeros() -> cons(0(), zeros())
, a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros^#() -> c_6()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
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(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)}
Weak Rules:
{ a__zeros() -> cons(0(), zeros())
, a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros^#() -> c_6()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__zeros_0() -> 4
, a__zeros_1() -> 4
, a__zeros_1() -> 5
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, zeros_0() -> 2
, zeros_0() -> 4
, zeros_1() -> 2
, zeros_1() -> 4
, zeros_1() -> 5
, a__tail_1(5) -> 4
, a__tail_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 4
, mark_1(2) -> 5
, tail_0(2) -> 2
, tail_1(5) -> 4
, tail_1(5) -> 5
, a__zeros^#_0() -> 1
, a__zeros^#_1() -> 7
, a__tail^#_0(2) -> 1
, a__tail^#_0(4) -> 3
, a__tail^#_1(5) -> 6
, c_1_0(1) -> 1
, c_1_1(8) -> 3
, c_1_1(8) -> 6
, mark^#_0(2) -> 1
, mark^#_1(2) -> 8
, c_2_0(1) -> 1
, c_2_1(7) -> 1
, c_2_1(7) -> 8
, c_3_0(3) -> 1
, c_3_1(6) -> 1
, c_3_1(6) -> 8
, c_4_0(1) -> 1
, c_4_1(8) -> 8
, c_6_0() -> 1
, c_6_1() -> 7}
2) { a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros^#() -> c_0()}
The usable rules for this path are the following:
{ mark(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(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(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(X)
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__zeros^#() -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__zeros() -> zeros()}
and weakly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__zeros() -> zeros()}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(zeros()) -> c_2(a__zeros^#())}
and weakly orienting the rules
{ a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(zeros()) -> c_2(a__zeros^#())}
Details:
Interpretation Functions:
a__zeros() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [8]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [3]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__zeros^#() -> c_0()}
and weakly orienting the rules
{ mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__zeros^#() -> c_0()}
Details:
Interpretation Functions:
a__zeros() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [8]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [8]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [3]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [7]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
and weakly orienting the rules
{ a__zeros^#() -> c_0()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
Details:
Interpretation Functions:
a__zeros() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
zeros() = [8]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [8]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__tail(X) -> tail(X)}
and weakly orienting the rules
{ mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros^#() -> c_0()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__tail(X) -> tail(X)}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [0]
zeros() = [1]
a__tail(x1) = [1] x1 + [8]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__zeros() -> cons(0(), zeros())}
and weakly orienting the rules
{ a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros^#() -> c_0()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__zeros() -> cons(0(), zeros())}
Details:
Interpretation Functions:
a__zeros() = [4]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
zeros() = [2]
a__tail(x1) = [1] x1 + [7]
mark(x1) = [1] x1 + [2]
tail(x1) = [1] x1 + [5]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [5]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [5]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [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(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)}
Weak Rules:
{ a__zeros() -> cons(0(), zeros())
, a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros^#() -> c_0()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
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(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)}
Weak Rules:
{ a__zeros() -> cons(0(), zeros())
, a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros^#() -> c_0()
, mark^#(zeros()) -> c_2(a__zeros^#())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__zeros_0() -> 4
, a__zeros_1() -> 4
, a__zeros_1() -> 5
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, zeros_0() -> 2
, zeros_0() -> 4
, zeros_1() -> 2
, zeros_1() -> 4
, zeros_1() -> 5
, a__tail_1(5) -> 4
, a__tail_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 4
, mark_1(2) -> 5
, tail_0(2) -> 2
, tail_1(5) -> 4
, tail_1(5) -> 5
, a__zeros^#_0() -> 1
, a__zeros^#_1() -> 7
, c_0_0() -> 1
, c_0_1() -> 7
, a__tail^#_0(2) -> 1
, a__tail^#_0(4) -> 3
, a__tail^#_1(5) -> 6
, c_1_0(1) -> 1
, c_1_1(8) -> 3
, c_1_1(8) -> 6
, mark^#_0(2) -> 1
, mark^#_1(2) -> 8
, c_2_0(1) -> 1
, c_2_1(7) -> 1
, c_2_1(7) -> 8
, c_3_0(3) -> 1
, c_3_1(6) -> 1
, c_3_1(6) -> 8
, c_4_0(1) -> 1
, c_4_1(8) -> 8}
3) { a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
The usable rules for this path are the following:
{ mark(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(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(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(X)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()}
and weakly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
and weakly orienting the rules
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
Details:
Interpretation Functions:
a__zeros() = [8]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [1]
zeros() = [7]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [9]
mark^#(x1) = [1] x1 + [4]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
and weakly orienting the rules
{ mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
Interpretation Functions:
a__zeros() = [7]
cons(x1, x2) = [1] x1 + [1] x2 + [6]
0() = [0]
zeros() = [1]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [6]
tail(x1) = [1] x1 + [13]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__tail(cons(X, XS)) -> mark(XS)}
and weakly orienting the rules
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__tail(cons(X, XS)) -> mark(XS)}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [1]
a__tail(x1) = [1] x1 + [9]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [15]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [5]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__tail(X) -> tail(X)}
Weak Rules:
{ a__tail(cons(X, XS)) -> mark(XS)
, mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, 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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__tail(X) -> tail(X)}
Weak Rules:
{ a__tail(cons(X, XS)) -> mark(XS)
, mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__zeros_0() -> 4
, a__zeros_1() -> 4
, a__zeros_1() -> 5
, a__zeros_2() -> 4
, a__zeros_2() -> 5
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, cons_2(8, 9) -> 4
, cons_2(8, 9) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, 0_2() -> 8
, zeros_0() -> 2
, zeros_0() -> 4
, zeros_1() -> 2
, zeros_1() -> 4
, zeros_1() -> 5
, zeros_2() -> 4
, zeros_2() -> 5
, zeros_2() -> 9
, a__tail_0(4) -> 4
, a__tail_1(5) -> 4
, a__tail_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 4
, mark_1(2) -> 5
, mark_1(9) -> 4
, mark_2(9) -> 4
, mark_2(9) -> 5
, tail_0(2) -> 2
, tail_1(4) -> 4
, tail_2(5) -> 4
, tail_2(5) -> 5
, a__tail^#_0(2) -> 1
, a__tail^#_0(4) -> 3
, a__tail^#_1(5) -> 7
, c_1_0(1) -> 1
, c_1_1(6) -> 3
, c_1_1(6) -> 7
, c_1_2(10) -> 7
, mark^#_0(2) -> 1
, mark^#_1(2) -> 6
, mark^#_1(9) -> 6
, mark^#_2(9) -> 10
, c_3_0(3) -> 1
, c_3_1(7) -> 1
, c_3_1(7) -> 6
, c_4_0(1) -> 1
, c_4_1(6) -> 6}
4) { a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__tail^#(X) -> c_7()}
The usable rules for this path are the following:
{ mark(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(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(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(X)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__tail^#(X) -> c_7()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()}
and weakly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [7]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
and weakly orienting the rules
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__tail(cons(X, XS)) -> mark(XS)
, a__tail(X) -> tail(X)}
and weakly orienting the rules
{ mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__tail(cons(X, XS)) -> mark(XS)
, a__tail(X) -> tail(X)}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [3]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
and weakly orienting the rules
{ a__tail(cons(X, XS)) -> mark(XS)
, a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
Interpretation Functions:
a__zeros() = [7]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
zeros() = [6]
a__tail(x1) = [1] x1 + [9]
mark(x1) = [1] x1 + [10]
tail(x1) = [1] x1 + [9]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [6]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [15]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))}
and weakly orienting the rules
{ mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))}
Details:
Interpretation Functions:
a__zeros() = [10]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [6]
zeros() = [2]
a__tail(x1) = [1] x1 + [13]
mark(x1) = [1] x1 + [10]
tail(x1) = [1] x1 + [13]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [6]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [3]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [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(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)}
Weak Rules:
{ a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
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(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)}
Weak Rules:
{ a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__tail(cons(X, XS)) -> mark(XS)
, a__tail(X) -> tail(X)
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, a__tail^#(X) -> c_7()}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a__zeros_0() -> 4
, a__zeros_1() -> 4
, a__zeros_1() -> 5
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, zeros_0() -> 2
, zeros_0() -> 4
, zeros_1() -> 2
, zeros_1() -> 4
, zeros_1() -> 5
, a__tail_1(5) -> 4
, a__tail_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 4
, mark_1(2) -> 5
, tail_0(2) -> 2
, tail_1(5) -> 4
, tail_1(5) -> 5
, a__tail^#_0(2) -> 1
, a__tail^#_0(4) -> 3
, a__tail^#_1(5) -> 7
, c_1_0(1) -> 1
, c_1_1(6) -> 3
, c_1_1(6) -> 7
, mark^#_0(2) -> 1
, mark^#_1(2) -> 6
, c_3_0(3) -> 1
, c_3_1(7) -> 1
, c_3_1(7) -> 6
, c_4_0(1) -> 1
, c_4_1(6) -> 6
, c_7_0() -> 1
, c_7_0() -> 3
, c_7_1() -> 7}
5) { a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(zeros()) -> c_2(a__zeros^#())}
The usable rules for this path are the following:
{ mark(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(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(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(X)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(zeros()) -> c_2(a__zeros^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, mark^#(zeros()) -> c_2(a__zeros^#())}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, mark^#(zeros()) -> c_2(a__zeros^#())}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()}
and weakly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, mark^#(zeros()) -> c_2(a__zeros^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
and weakly orienting the rules
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, mark^#(zeros()) -> c_2(a__zeros^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
and weakly orienting the rules
{ mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, mark^#(zeros()) -> c_2(a__zeros^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
Interpretation Functions:
a__zeros() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [1]
zeros() = [0]
a__tail(x1) = [1] x1 + [2]
mark(x1) = [1] x1 + [4]
tail(x1) = [1] x1 + [13]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__tail(cons(X, XS)) -> mark(XS)}
and weakly orienting the rules
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, mark^#(zeros()) -> c_2(a__zeros^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__tail(cons(X, XS)) -> mark(XS)}
Details:
Interpretation Functions:
a__zeros() = [4]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [1]
zeros() = [3]
a__tail(x1) = [1] x1 + [9]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [15]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [14]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__tail(X) -> tail(X)}
Weak Rules:
{ a__tail(cons(X, XS)) -> mark(XS)
, mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, mark^#(zeros()) -> c_2(a__zeros^#())}
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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__tail(X) -> tail(X)}
Weak Rules:
{ a__tail(cons(X, XS)) -> mark(XS)
, mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()
, mark^#(zeros()) -> c_2(a__zeros^#())}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__zeros_0() -> 4
, a__zeros_1() -> 4
, a__zeros_1() -> 5
, a__zeros_2() -> 4
, a__zeros_2() -> 5
, 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
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, 0_2() -> 9
, zeros_0() -> 2
, zeros_0() -> 4
, zeros_1() -> 2
, zeros_1() -> 4
, zeros_1() -> 5
, zeros_2() -> 4
, zeros_2() -> 5
, zeros_2() -> 10
, a__tail_0(4) -> 4
, a__tail_1(5) -> 4
, a__tail_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 4
, mark_1(2) -> 5
, mark_1(10) -> 4
, mark_2(10) -> 4
, mark_2(10) -> 5
, tail_0(2) -> 2
, tail_1(4) -> 4
, tail_2(5) -> 4
, tail_2(5) -> 5
, a__zeros^#_0() -> 1
, a__zeros^#_1() -> 8
, a__zeros^#_2() -> 12
, a__tail^#_0(2) -> 1
, a__tail^#_0(4) -> 3
, a__tail^#_1(5) -> 7
, c_1_0(1) -> 1
, c_1_1(6) -> 3
, c_1_1(6) -> 7
, c_1_2(11) -> 7
, mark^#_0(2) -> 1
, mark^#_1(2) -> 6
, mark^#_1(10) -> 6
, mark^#_2(10) -> 11
, c_2_0(1) -> 1
, c_2_1(8) -> 1
, c_2_1(8) -> 6
, c_2_2(12) -> 6
, c_2_2(12) -> 11
, c_3_0(3) -> 1
, c_3_1(7) -> 1
, c_3_1(7) -> 6
, c_4_0(1) -> 1
, c_4_1(6) -> 6}
6) { a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(0()) -> c_5()}
The usable rules for this path are the following:
{ mark(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(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(zeros()) -> a__zeros()
, mark(tail(X)) -> a__tail(mark(X))
, mark(cons(X1, X2)) -> cons(mark(X1), X2)
, mark(0()) -> 0()
, a__zeros() -> cons(0(), zeros())
, a__tail(cons(X, XS)) -> mark(XS)
, a__zeros() -> zeros()
, a__tail(X) -> tail(X)
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, mark^#(0()) -> c_5()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(0()) -> c_5()}
and weakly orienting the rules
{ mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(0()) -> c_5()}
Details:
Interpretation Functions:
a__zeros() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [0]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [0]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [5]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
and weakly orienting the rules
{ a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(0()) -> c_5()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))}
Details:
Interpretation Functions:
a__zeros() = [14]
cons(x1, x2) = [1] x1 + [1] x2 + [13]
0() = [0]
zeros() = [1]
a__tail(x1) = [1] x1 + [2]
mark(x1) = [1] x1 + [15]
tail(x1) = [1] x1 + [15]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__tail(cons(X, XS)) -> mark(XS)}
and weakly orienting the rules
{ mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(0()) -> c_5()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__tail(cons(X, XS)) -> mark(XS)}
Details:
Interpretation Functions:
a__zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
zeros() = [0]
a__tail(x1) = [1] x1 + [9]
mark(x1) = [1] x1 + [1]
tail(x1) = [1] x1 + [15]
a__zeros^#() = [0]
c_0() = [0]
a__tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6() = [0]
c_7() = [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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__tail(X) -> tail(X)}
Weak Rules:
{ a__tail(cons(X, XS)) -> mark(XS)
, mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(0()) -> c_5()
, mark(zeros()) -> a__zeros()
, 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(cons(X1, X2)) -> cons(mark(X1), X2)
, a__tail(X) -> tail(X)}
Weak Rules:
{ a__tail(cons(X, XS)) -> mark(XS)
, mark(tail(X)) -> a__tail(mark(X))
, a__tail^#(cons(X, XS)) -> c_1(mark^#(XS))
, mark^#(cons(X1, X2)) -> c_4(mark^#(X1))
, a__zeros() -> cons(0(), zeros())
, a__zeros() -> zeros()
, mark^#(tail(X)) -> c_3(a__tail^#(mark(X)))
, mark^#(0()) -> c_5()
, mark(zeros()) -> a__zeros()
, mark(0()) -> 0()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a__zeros_0() -> 4
, a__zeros_1() -> 4
, a__zeros_1() -> 5
, a__zeros_2() -> 4
, a__zeros_2() -> 5
, cons_0(2, 2) -> 2
, cons_1(5, 2) -> 4
, cons_1(5, 2) -> 5
, cons_2(8, 9) -> 4
, cons_2(8, 9) -> 5
, 0_0() -> 2
, 0_0() -> 4
, 0_1() -> 4
, 0_1() -> 5
, 0_2() -> 8
, zeros_0() -> 2
, zeros_0() -> 4
, zeros_1() -> 2
, zeros_1() -> 4
, zeros_1() -> 5
, zeros_2() -> 4
, zeros_2() -> 5
, zeros_2() -> 9
, a__tail_0(4) -> 4
, a__tail_1(5) -> 4
, a__tail_1(5) -> 5
, mark_0(2) -> 4
, mark_1(2) -> 4
, mark_1(2) -> 5
, mark_1(9) -> 4
, mark_2(9) -> 4
, mark_2(9) -> 5
, tail_0(2) -> 2
, tail_1(4) -> 4
, tail_2(5) -> 4
, tail_2(5) -> 5
, a__tail^#_0(2) -> 1
, a__tail^#_0(4) -> 3
, a__tail^#_1(5) -> 7
, c_1_0(1) -> 1
, c_1_1(6) -> 3
, c_1_1(6) -> 7
, c_1_2(10) -> 7
, mark^#_0(2) -> 1
, mark^#_1(2) -> 6
, mark^#_1(9) -> 6
, mark^#_2(9) -> 10
, c_3_0(3) -> 1
, c_3_1(7) -> 1
, c_3_1(7) -> 6
, c_4_0(1) -> 1
, c_4_1(6) -> 6
, c_5_0() -> 1
, c_5_1() -> 6}