'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, top(mark(X)) -> top(proper(X))
, top(ok(X)) -> top(active(X))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))
, active^#(__(X, nil())) -> c_1()
, active^#(__(nil(), X)) -> c_2()
, active^#(and(tt(), X)) -> c_3()
, active^#(isNePal(__(I, __(P, I)))) -> c_4()
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, proper^#(nil()) -> c_14()
, proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, proper^#(tt()) -> c_16()
, proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, top^#(mark(X)) -> c_21(top^#(proper(X)))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
The usable rules are:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))}
The estimated dependency graph contains the following edges:
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
==> {__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
==> {__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
==> {__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
{active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
==> {__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
{active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
==> {__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
{active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
==> {__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
{active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
==> {__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
{active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
==> {__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
{active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
==> {__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
{active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))}
==> {and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
{active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))}
==> {and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
{active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))}
==> {isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
{active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))}
==> {isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
{__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
==> {__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
{__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
==> {__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
{__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
==> {__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
{__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
==> {__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
{__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
==> {__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
{__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
==> {__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
{and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
==> {and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
{and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
==> {and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
{isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
==> {isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
{isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
==> {isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
{proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
==> {__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
{proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
==> {__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
{proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
==> {__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
{proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))}
==> {and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
{proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))}
==> {and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
{proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))}
==> {isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
{proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))}
==> {isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
{__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
==> {__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
{__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
==> {__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
{__^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
==> {__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
{and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
==> {and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
{and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
==> {and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
{isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
==> {isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
{isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
==> {isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
{top^#(mark(X)) -> c_21(top^#(proper(X)))}
==> {top^#(ok(X)) -> c_22(top^#(active(X)))}
{top^#(mark(X)) -> c_21(top^#(proper(X)))}
==> {top^#(mark(X)) -> c_21(top^#(proper(X)))}
{top^#(ok(X)) -> c_22(top^#(active(X)))}
==> {top^#(ok(X)) -> c_22(top^#(active(X)))}
{top^#(ok(X)) -> c_22(top^#(active(X)))}
==> {top^#(mark(X)) -> c_21(top^#(proper(X)))}
We consider the following path(s):
1) { top^#(mark(X)) -> c_21(top^#(proper(X)))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, top^#(mark(X)) -> c_21(top^#(proper(X)))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [5]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_21(x1) = [1] x1 + [0]
c_22(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [2]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [7]
c_21(x1) = [1] x1 + [0]
c_22(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(and(tt(), X)) -> mark(X)}
and weakly orienting the rules
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(and(tt(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [4]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [15]
c_21(x1) = [1] x1 + [4]
c_22(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
and weakly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [4]
proper(x1) = [1] x1 + [8]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_21(x1) = [1] x1 + [8]
c_22(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
and weakly orienting the rules
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [3]
nil() = [3]
and(x1, x2) = [1] x1 + [1] x2 + [8]
tt() = [6]
isNePal(x1) = [1] x1 + [9]
proper(x1) = [1] x1 + [8]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [8]
c_21(x1) = [1] x1 + [1]
c_22(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{top^#(mark(X)) -> c_21(top^#(proper(X)))}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{top^#(mark(X)) -> c_21(top^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [4]
nil() = [4]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [8]
isNePal(x1) = [1] x1 + [12]
proper(x1) = [1] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [1] x1 + [0]
c_21(x1) = [1] x1 + [0]
c_22(x1) = [1] x1 + [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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ top^#(mark(X)) -> c_21(top^#(proper(X)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ top^#(mark(X)) -> c_21(top^#(proper(X)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, top^#(ok(X)) -> c_22(top^#(active(X)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ active_0(2) -> 6
, mark_0(2) -> 2
, nil_0() -> 2
, tt_0() -> 2
, proper_0(2) -> 4
, ok_0(2) -> 2
, ok_0(2) -> 4
, top^#_0(2) -> 1
, top^#_0(4) -> 3
, top^#_0(6) -> 5
, c_21_0(3) -> 1
, c_22_0(5) -> 1
, c_22_0(5) -> 3}
2) { active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [3]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
and weakly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
and weakly orienting the rules
{ active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [2]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
and weakly orienting the rules
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [9]
mark(x1) = [1] x1 + [0]
nil() = [8]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [2]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(2) -> 2
, nil_0() -> 2
, tt_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, __^#_0(2, 2) -> 1
, c_9_0(1) -> 1
, c_10_0(1) -> 1
, c_18_0(1) -> 1}
3) { active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [3]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
and weakly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
and weakly orienting the rules
{ active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [2]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
and weakly orienting the rules
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [9]
mark(x1) = [1] x1 + [0]
nil() = [8]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [2]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(2) -> 2
, nil_0() -> 2
, tt_0() -> 2
, ok_0(2) -> 2
, active^#_0(2) -> 1
, __^#_0(2, 2) -> 1
, c_9_0(1) -> 1
, c_10_0(1) -> 1
, c_18_0(1) -> 1}
4) { active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [2]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [15]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [1]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(and(tt(), X)) -> mark(X)
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [3]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [1]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
and weakly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [4]
tt() = [0]
isNePal(x1) = [1] x1 + [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [1]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
and weakly orienting the rules
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(and(tt(), X)) -> mark(X)
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
and weakly orienting the rules
{ and^#(mark(X1), X2) -> c_11(and^#(X1, X2))
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(and(tt(), X)) -> mark(X)
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [8]
nil() = [10]
and(x1, x2) = [1] x1 + [1] x2 + [8]
tt() = [0]
isNePal(x1) = [1] x1 + [7]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(and(tt(), X)) -> mark(X)
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [4]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [2]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(and(tt(), X)) -> mark(X)
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(and(tt(), X)) -> mark(X)
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, active^#_0(3) -> 11
, active^#_0(4) -> 11
, active^#_0(6) -> 11
, active^#_0(9) -> 11
, and^#_0(3, 3) -> 21
, and^#_0(3, 4) -> 21
, and^#_0(3, 6) -> 21
, and^#_0(3, 9) -> 21
, and^#_0(4, 3) -> 21
, and^#_0(4, 4) -> 21
, and^#_0(4, 6) -> 21
, and^#_0(4, 9) -> 21
, and^#_0(6, 3) -> 21
, and^#_0(6, 4) -> 21
, and^#_0(6, 6) -> 21
, and^#_0(6, 9) -> 21
, and^#_0(9, 3) -> 21
, and^#_0(9, 4) -> 21
, and^#_0(9, 6) -> 21
, and^#_0(9, 9) -> 21
, c_11_0(21) -> 21
, c_19_0(21) -> 21}
5) { active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [1]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
and weakly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [1]
isNePal^#(x1) = [1] x1 + [1]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
and weakly orienting the rules
{ isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [10]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [1]
isNePal^#(x1) = [1] x1 + [3]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))}
and weakly orienting the rules
{ isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [12]
mark(x1) = [1] x1 + [0]
nil() = [8]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [8]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
isNePal^#(x1) = [1] x1 + [5]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [2]
isNePal^#(x1) = [1] x1 + [5]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, active(and(tt(), X)) -> mark(X)
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, active^#_0(3) -> 11
, active^#_0(4) -> 11
, active^#_0(6) -> 11
, active^#_0(9) -> 11
, isNePal^#_0(3) -> 23
, isNePal^#_0(4) -> 23
, isNePal^#_0(6) -> 23
, isNePal^#_0(9) -> 23
, c_12_0(23) -> 23
, c_20_0(23) -> 23}
6) {active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [7]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [12]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [5]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(and(tt(), X)) -> mark(X)}
and weakly orienting the rules
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(and(tt(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [8]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [2]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
and weakly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [15]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [8]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [4]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_6(__^#(X1, active(X2)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, active^#_0(3) -> 11
, active^#_0(4) -> 11
, active^#_0(6) -> 11
, active^#_0(9) -> 11
, __^#_0(3, 3) -> 13
, __^#_0(3, 4) -> 13
, __^#_0(3, 6) -> 13
, __^#_0(3, 9) -> 13
, __^#_0(4, 3) -> 13
, __^#_0(4, 4) -> 13
, __^#_0(4, 6) -> 13
, __^#_0(4, 9) -> 13
, __^#_0(6, 3) -> 13
, __^#_0(6, 4) -> 13
, __^#_0(6, 6) -> 13
, __^#_0(6, 9) -> 13
, __^#_0(9, 3) -> 13
, __^#_0(9, 4) -> 13
, __^#_0(9, 6) -> 13
, __^#_0(9, 9) -> 13}
7) {active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [7]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(isNePal(__(I, __(P, I)))) -> mark(tt())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [12]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [5]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(and(tt(), X)) -> mark(X)}
and weakly orienting the rules
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(and(tt(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [8]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [2]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
and weakly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [15]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [8]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [4]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active^#(__(X1, X2)) -> c_5(__^#(active(X1), X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, active^#_0(3) -> 11
, active^#_0(4) -> 11
, active^#_0(6) -> 11
, active^#_0(9) -> 11
, __^#_0(3, 3) -> 13
, __^#_0(3, 4) -> 13
, __^#_0(3, 6) -> 13
, __^#_0(3, 9) -> 13
, __^#_0(4, 3) -> 13
, __^#_0(4, 4) -> 13
, __^#_0(4, 6) -> 13
, __^#_0(4, 9) -> 13
, __^#_0(6, 3) -> 13
, __^#_0(6, 4) -> 13
, __^#_0(6, 6) -> 13
, __^#_0(6, 9) -> 13
, __^#_0(9, 3) -> 13
, __^#_0(9, 4) -> 13
, __^#_0(9, 6) -> 13
, __^#_0(9, 9) -> 13}
8) { proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
The usable rules for this path are the following:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [1]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
and weakly orienting the rules
{ proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{__^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [8]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [4]
c_10(x1) = [1] x1 + [8]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [2]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
and weakly orienting the rules
{ __^#(mark(X1), X2) -> c_9(__^#(X1, X2))
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{__^#(X1, mark(X2)) -> c_10(__^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [8]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [13]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [15]
c_13(x1) = [1] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
and weakly orienting the rules
{ __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [4]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [1]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(2) -> 2
, nil_0() -> 2
, tt_0() -> 2
, ok_0(2) -> 2
, __^#_0(2, 2) -> 1
, c_9_0(1) -> 1
, c_10_0(1) -> 1
, proper^#_0(2) -> 1
, c_18_0(1) -> 1}
9) {active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())}
and weakly orienting the rules
{ active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [4]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [12]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(and(tt(), X)) -> mark(X)}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(and(tt(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [15]
and(x1, x2) = [1] x1 + [1] x2 + [8]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [15]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
and weakly orienting the rules
{ active(and(tt(), X)) -> mark(X)
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [15]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(and(tt(), X)) -> mark(X)
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(and(tt(), X)) -> mark(X)
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(and(X1, X2)) -> c_7(and^#(active(X1), X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, active^#_0(3) -> 11
, active^#_0(4) -> 11
, active^#_0(6) -> 11
, active^#_0(9) -> 11
, and^#_0(3, 3) -> 21
, and^#_0(3, 4) -> 21
, and^#_0(3, 6) -> 21
, and^#_0(3, 9) -> 21
, and^#_0(4, 3) -> 21
, and^#_0(4, 4) -> 21
, and^#_0(4, 6) -> 21
, and^#_0(4, 9) -> 21
, and^#_0(6, 3) -> 21
, and^#_0(6, 4) -> 21
, and^#_0(6, 6) -> 21
, and^#_0(6, 9) -> 21
, and^#_0(9, 3) -> 21
, and^#_0(9, 4) -> 21
, and^#_0(9, 6) -> 21
, and^#_0(9, 9) -> 21}
10)
{active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))}
The usable rules for this path are the following:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active(and(tt(), X)) -> mark(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(and(tt(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [8]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{active(and(tt(), X)) -> mark(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [4]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [4]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active(and(tt(), X)) -> mark(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [2]
isNePal(x1) = [1] x1 + [6]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [10]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [7]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
and weakly orienting the rules
{ active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active(and(tt(), X)) -> mark(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [15]
and(x1, x2) = [1] x1 + [1] x2 + [1]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
and weakly orienting the rules
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active(and(tt(), X)) -> mark(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [1] x1 + [1]
__(x1, x2) = [1] x1 + [1] x2 + [11]
mark(x1) = [1] x1 + [0]
nil() = [5]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active(and(tt(), X)) -> mark(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:
{ active(__(X1, X2)) -> __(active(X1), X2)
, active(__(X1, X2)) -> __(X1, active(X2))
, active(and(X1, X2)) -> and(active(X1), X2)
, active(isNePal(X)) -> isNePal(active(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, active^#(isNePal(X)) -> c_8(isNePal^#(active(X)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, active(and(tt(), X)) -> mark(X)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, active^#_0(3) -> 11
, active^#_0(4) -> 11
, active^#_0(6) -> 11
, active^#_0(9) -> 11
, isNePal^#_0(3) -> 23
, isNePal^#_0(4) -> 23
, isNePal^#_0(6) -> 23
, isNePal^#_0(9) -> 23}
11)
{ proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
The usable rules for this path are the following:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [1] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{isNePal^#(ok(X)) -> c_20(isNePal^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [1] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))}
and weakly orienting the rules
{ isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [1]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [1] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
and weakly orienting the rules
{ proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{isNePal^#(mark(X)) -> c_12(isNePal^#(X))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [8]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [1] x1 + [7]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [1]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
and weakly orienting the rules
{ isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [15]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [8]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [1] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [1] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, isNePal^#(mark(X)) -> c_12(isNePal^#(X))
, proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, isNePal^#(ok(X)) -> c_20(isNePal^#(X))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, isNePal^#_0(3) -> 23
, isNePal^#_0(4) -> 23
, isNePal^#_0(6) -> 23
, isNePal^#_0(9) -> 23
, c_12_0(23) -> 23
, proper^#_0(3) -> 28
, proper^#_0(4) -> 28
, proper^#_0(6) -> 28
, proper^#_0(9) -> 28
, c_20_0(23) -> 23}
12)
{ proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
The usable rules for this path are the following:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [1] x1 + [1]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [1]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [12]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [3]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [1] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [1]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
and weakly orienting the rules
{ proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{and^#(mark(X1), X2) -> c_11(and^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [8]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [1] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
and weakly orienting the rules
{ and^#(mark(X1), X2) -> c_11(and^#(X1, X2))
, proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [8]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [8]
proper(x1) = [1] x1 + [2]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [2]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [1] x1 + [3]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [1] x1 + [1]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [1] x1 + [1]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))
, proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, and^#(mark(X1), X2) -> c_11(and^#(X1, X2))
, proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, and^#(ok(X1), ok(X2)) -> c_19(and^#(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, and^#_0(3, 3) -> 21
, and^#_0(3, 4) -> 21
, and^#_0(3, 6) -> 21
, and^#_0(3, 9) -> 21
, and^#_0(4, 3) -> 21
, and^#_0(4, 4) -> 21
, and^#_0(4, 6) -> 21
, and^#_0(4, 9) -> 21
, and^#_0(6, 3) -> 21
, and^#_0(6, 4) -> 21
, and^#_0(6, 6) -> 21
, and^#_0(6, 9) -> 21
, and^#_0(9, 3) -> 21
, and^#_0(9, 4) -> 21
, and^#_0(9, 6) -> 21
, and^#_0(9, 9) -> 21
, c_11_0(21) -> 21
, proper^#_0(3) -> 28
, proper^#_0(4) -> 28
, proper^#_0(6) -> 28
, proper^#_0(9) -> 28
, c_19_0(21) -> 21}
13)
{proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
The usable rules for this path are the following:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [1] x1 + [1]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [2]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [1] x1 + [1]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
and weakly orienting the rules
{ proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [2]
mark(x1) = [1] x1 + [0]
nil() = [8]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [1]
proper(x1) = [1] x1 + [8]
ok(x1) = [1] x1 + [7]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [15]
c_13(x1) = [1] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, proper^#(__(X1, X2)) -> c_13(__^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, __^#_0(3, 3) -> 13
, __^#_0(3, 4) -> 13
, __^#_0(3, 6) -> 13
, __^#_0(3, 9) -> 13
, __^#_0(4, 3) -> 13
, __^#_0(4, 4) -> 13
, __^#_0(4, 6) -> 13
, __^#_0(4, 9) -> 13
, __^#_0(6, 3) -> 13
, __^#_0(6, 4) -> 13
, __^#_0(6, 6) -> 13
, __^#_0(6, 9) -> 13
, __^#_0(9, 3) -> 13
, __^#_0(9, 4) -> 13
, __^#_0(9, 6) -> 13
, __^#_0(9, 9) -> 13
, proper^#_0(3) -> 28
, proper^#_0(4) -> 28
, proper^#_0(6) -> 28
, proper^#_0(9) -> 28}
14)
{proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))}
The usable rules for this path are the following:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [1] x1 + [1]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [1]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [2]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [1] x1 + [1]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
and weakly orienting the rules
{ proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [8]
and(x1, x2) = [1] x1 + [1] x2 + [2]
tt() = [0]
isNePal(x1) = [1] x1 + [1]
proper(x1) = [1] x1 + [8]
ok(x1) = [1] x1 + [7]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [15]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [1] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, proper^#(and(X1, X2)) -> c_15(and^#(proper(X1), proper(X2)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, and^#_0(3, 3) -> 21
, and^#_0(3, 4) -> 21
, and^#_0(3, 6) -> 21
, and^#_0(3, 9) -> 21
, and^#_0(4, 3) -> 21
, and^#_0(4, 4) -> 21
, and^#_0(4, 6) -> 21
, and^#_0(4, 9) -> 21
, and^#_0(6, 3) -> 21
, and^#_0(6, 4) -> 21
, and^#_0(6, 6) -> 21
, and^#_0(6, 9) -> 21
, and^#_0(9, 3) -> 21
, and^#_0(9, 4) -> 21
, and^#_0(9, 6) -> 21
, and^#_0(9, 9) -> 21
, proper^#_0(3) -> 28
, proper^#_0(4) -> 28
, proper^#_0(6) -> 28
, proper^#_0(9) -> 28}
15)
{proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))}
The usable rules for this path are the following:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(nil()) -> ok(nil())
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(tt()) -> ok(tt())
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))
, isNePal(ok(X)) -> ok(isNePal(X))
, proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [1] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [1] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
and weakly orienting the rules
{ proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [1] x1 + [0]
nil() = [15]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [1] x1 + [1]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [1] x1 + [2]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [9]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [1] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
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:
{ proper(__(X1, X2)) -> __(proper(X1), proper(X2))
, proper(and(X1, X2)) -> and(proper(X1), proper(X2))
, proper(isNePal(X)) -> isNePal(proper(X))
, __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, and(mark(X1), X2) -> mark(and(X1, X2))
, isNePal(mark(X)) -> mark(isNePal(X))
, isNePal(ok(X)) -> ok(isNePal(X))}
Weak Rules:
{ proper(nil()) -> ok(nil())
, proper(tt()) -> ok(tt())
, proper^#(isNePal(X)) -> c_17(isNePal^#(proper(X)))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, and(ok(X1), ok(X2)) -> ok(and(X1, X2))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(4) -> 3
, mark_0(6) -> 3
, mark_0(9) -> 3
, nil_0() -> 4
, tt_0() -> 6
, ok_0(3) -> 9
, ok_0(4) -> 9
, ok_0(6) -> 9
, ok_0(9) -> 9
, isNePal^#_0(3) -> 23
, isNePal^#_0(4) -> 23
, isNePal^#_0(6) -> 23
, isNePal^#_0(9) -> 23
, proper^#_0(3) -> 28
, proper^#_0(4) -> 28
, proper^#_0(6) -> 28
, proper^#_0(9) -> 28}
16)
{ active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
The usable rules for this path are the following:
{ __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))}
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(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [1]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [1]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
and weakly orienting the rules
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [1]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [2]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [1]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
and weakly orienting the rules
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [1]
mark(x1) = [1] x1 + [8]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [1]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [4]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [1]
c_10(x1) = [1] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [1] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))}
Weak Rules:
{ __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
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(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))}
Weak Rules:
{ __^#(X1, mark(X2)) -> c_10(__^#(X1, X2))
, __^#(mark(X1), X2) -> c_9(__^#(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, __^#(ok(X1), ok(X2)) -> c_18(__^#(X1, X2))
, active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(9) -> 3
, ok_0(3) -> 9
, ok_0(9) -> 9
, active^#_0(3) -> 11
, active^#_0(9) -> 11
, __^#_0(3, 3) -> 13
, __^#_0(3, 9) -> 13
, __^#_0(9, 3) -> 13
, __^#_0(9, 9) -> 13
, c_9_0(13) -> 13
, c_10_0(13) -> 13
, c_18_0(13) -> 13}
17)
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
The usable rules for this path are the following:
{ __(mark(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))}
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(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))
, __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [1]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{__(ok(X1), ok(X2)) -> ok(__(X1, X2))}
and weakly orienting the rules
{active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{__(ok(X1), ok(X2)) -> ok(__(X1, X2))}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [1]
mark(x1) = [1] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [1] x1 + [8]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
__^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [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(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))}
Weak Rules:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
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(X1), X2) -> mark(__(X1, X2))
, __(X1, mark(X2)) -> mark(__(X1, X2))}
Weak Rules:
{ __(ok(X1), ok(X2)) -> ok(__(X1, X2))
, active^#(__(__(X, Y), Z)) -> c_0(__^#(X, __(Y, Z)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ mark_0(3) -> 3
, mark_0(9) -> 3
, ok_0(3) -> 9
, ok_0(9) -> 9
, active^#_0(3) -> 11
, active^#_0(9) -> 11
, __^#_0(3, 3) -> 13
, __^#_0(3, 9) -> 13
, __^#_0(9, 3) -> 13
, __^#_0(9, 9) -> 13}
18)
{active^#(__(nil(), X)) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {active^#(__(nil(), X)) -> c_2()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(__(nil(), X)) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(__(nil(), X)) -> c_2()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {active^#(__(nil(), X)) -> c_2()}
Details:
The given problem does not contain any strict rules
19)
{active^#(isNePal(__(I, __(P, I)))) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {active^#(isNePal(__(I, __(P, I)))) -> c_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(isNePal(__(I, __(P, I)))) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(isNePal(__(I, __(P, I)))) -> c_4()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [1] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {active^#(isNePal(__(I, __(P, I)))) -> c_4()}
Details:
The given problem does not contain any strict rules
20)
{active^#(__(X, nil())) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {active^#(__(X, nil())) -> c_1()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(__(X, nil())) -> c_1()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(__(X, nil())) -> c_1()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [1] x1 + [1] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {active^#(__(X, nil())) -> c_1()}
Details:
The given problem does not contain any strict rules
21)
{active^#(and(tt(), X)) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {active^#(and(tt(), X)) -> c_3()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{active^#(and(tt(), X)) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{active^#(and(tt(), X)) -> c_3()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [1] x1 + [1] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {active^#(and(tt(), X)) -> c_3()}
Details:
The given problem does not contain any strict rules
22)
{proper^#(tt()) -> c_16()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {proper^#(tt()) -> c_16()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(tt()) -> c_16()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(tt()) -> c_16()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {proper^#(tt()) -> c_16()}
Details:
The given problem does not contain any strict rules
23)
{proper^#(nil()) -> c_14()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {proper^#(nil()) -> c_14()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{proper^#(nil()) -> c_14()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{proper^#(nil()) -> c_14()}
Details:
Interpretation Functions:
active(x1) = [0] x1 + [0]
__(x1, x2) = [0] x1 + [0] x2 + [0]
mark(x1) = [0] x1 + [0]
nil() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isNePal(x1) = [0] x1 + [0]
proper(x1) = [0] x1 + [0]
ok(x1) = [0] x1 + [0]
top(x1) = [0] x1 + [0]
active^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
__^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
isNePal^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
proper^#(x1) = [1] x1 + [1]
c_13(x1) = [0] x1 + [0]
c_14() = [0]
c_15(x1) = [0] x1 + [0]
c_16() = [0]
c_17(x1) = [0] x1 + [0]
c_18(x1) = [0] x1 + [0]
c_19(x1) = [0] x1 + [0]
c_20(x1) = [0] x1 + [0]
top^#(x1) = [0] x1 + [0]
c_21(x1) = [0] x1 + [0]
c_22(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {proper^#(nil()) -> c_14()}
Details:
The given problem does not contain any strict rules