'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ fst(0(), Z) -> nil()
, fst(s(X), cons(Y, Z)) ->
cons(Y, n__fst(activate(X), activate(Z)))
, from(X) -> cons(X, n__from(s(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(n__add(activate(X), Y))
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(n__len(activate(Z)))
, fst(X1, X2) -> n__fst(X1, X2)
, from(X) -> n__from(X)
, add(X1, X2) -> n__add(X1, X2)
, len(X) -> n__len(X)
, activate(n__fst(X1, X2)) -> fst(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(n__add(X1, X2)) -> add(X1, X2)
, activate(n__len(X)) -> len(X)
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ fst^#(0(), Z) -> c_0()
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, from^#(X) -> c_2()
, add^#(0(), X) -> c_3()
, add^#(s(X), Y) -> c_4(activate^#(X))
, len^#(nil()) -> c_5()
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, fst^#(X1, X2) -> c_7()
, from^#(X) -> c_8()
, add^#(X1, X2) -> c_9()
, len^#(X) -> c_10()
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, activate^#(n__from(X)) -> c_12(from^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, activate^#(n__len(X)) -> c_14(len^#(X))
, activate^#(X) -> c_15()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))}
==> {activate^#(n__len(X)) -> c_14(len^#(X))}
{fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))}
==> {activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
{fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))}
==> {activate^#(n__from(X)) -> c_12(from^#(X))}
{fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))}
==> {activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))}
{fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))}
==> {activate^#(X) -> c_15()}
{add^#(s(X), Y) -> c_4(activate^#(X))}
==> {activate^#(n__len(X)) -> c_14(len^#(X))}
{add^#(s(X), Y) -> c_4(activate^#(X))}
==> {activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
{add^#(s(X), Y) -> c_4(activate^#(X))}
==> {activate^#(n__from(X)) -> c_12(from^#(X))}
{add^#(s(X), Y) -> c_4(activate^#(X))}
==> {activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))}
{add^#(s(X), Y) -> c_4(activate^#(X))}
==> {activate^#(X) -> c_15()}
{len^#(cons(X, Z)) -> c_6(activate^#(Z))}
==> {activate^#(n__len(X)) -> c_14(len^#(X))}
{len^#(cons(X, Z)) -> c_6(activate^#(Z))}
==> {activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
{len^#(cons(X, Z)) -> c_6(activate^#(Z))}
==> {activate^#(n__from(X)) -> c_12(from^#(X))}
{len^#(cons(X, Z)) -> c_6(activate^#(Z))}
==> {activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))}
{len^#(cons(X, Z)) -> c_6(activate^#(Z))}
==> {activate^#(X) -> c_15()}
{activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))}
==> {fst^#(X1, X2) -> c_7()}
{activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))}
==> {fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))}
{activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))}
==> {fst^#(0(), Z) -> c_0()}
{activate^#(n__from(X)) -> c_12(from^#(X))}
==> {from^#(X) -> c_8()}
{activate^#(n__from(X)) -> c_12(from^#(X))}
==> {from^#(X) -> c_2()}
{activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
==> {add^#(X1, X2) -> c_9()}
{activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
==> {add^#(s(X), Y) -> c_4(activate^#(X))}
{activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
==> {add^#(0(), X) -> c_3()}
{activate^#(n__len(X)) -> c_14(len^#(X))}
==> {len^#(X) -> c_10()}
{activate^#(n__len(X)) -> c_14(len^#(X))}
==> {len^#(cons(X, Z)) -> c_6(activate^#(Z))}
{activate^#(n__len(X)) -> c_14(len^#(X))}
==> {len^#(nil()) -> c_5()}
We consider the following path(s):
1) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate^#(n__len(X)) -> c_14(len^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate^#(n__len(X)) -> c_14(len^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [2]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [2]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{add^#(s(X), Y) -> c_4(activate^#(X))}
and weakly orienting the rules
{ activate^#(n__len(X)) -> c_14(len^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{add^#(s(X), Y) -> c_4(activate^#(X))}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [12]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [9]
c_3() = [0]
c_4(x1) = [1] x1 + [3]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{len^#(cons(X, Z)) -> c_6(activate^#(Z))}
and weakly orienting the rules
{ add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__len(X)) -> c_14(len^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{len^#(cons(X, Z)) -> c_6(activate^#(Z))}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [8]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))}
and weakly orienting the rules
{ len^#(cons(X, Z)) -> c_6(activate^#(Z))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__len(X)) -> c_14(len^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [12]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [7]
c_14(x1) = [1] x1 + [0]
c_15() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))}
and weakly orienting the rules
{ activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__len(X)) -> c_14(len^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [8]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [0]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
len^#(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [1]
c_15() = [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:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__len(X)) -> c_14(len^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
2) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, activate^#(n__from(X)) -> c_12(from^#(X))
, from^#(X) -> c_8()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {from^#(X) -> c_8()}
Weak Rules:
{ activate^#(n__from(X)) -> c_12(from^#(X))
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{from^#(X) -> c_8()}
and weakly orienting the rules
{ activate^#(n__from(X)) -> c_12(from^#(X))
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{from^#(X) -> c_8()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [3]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [0]
fst^#(x1, x2) = [1] x1 + [1] x2 + [9]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [3]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [1] x1 + [1]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [3]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15() = [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:
{ from^#(X) -> c_8()
, activate^#(n__from(X)) -> c_12(from^#(X))
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
3) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, fst^#(0(), Z) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {fst^#(0(), Z) -> c_0()}
Weak Rules:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{fst^#(0(), Z) -> c_0()}
and weakly orienting the rules
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{fst^#(0(), Z) -> c_0()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [8]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [7]
c_15() = [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:
{ fst^#(0(), Z) -> c_0()
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
4) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, activate^#(n__from(X)) -> c_12(from^#(X))
, from^#(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:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {from^#(X) -> c_2()}
Weak Rules:
{ activate^#(n__from(X)) -> c_12(from^#(X))
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{from^#(X) -> c_2()}
and weakly orienting the rules
{ activate^#(n__from(X)) -> c_12(from^#(X))
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{from^#(X) -> c_2()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [3]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [0]
fst^#(x1, x2) = [1] x1 + [1] x2 + [9]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [3]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [1] x1 + [1]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [3]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [0]
c_15() = [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:
{ from^#(X) -> c_2()
, activate^#(n__from(X)) -> c_12(from^#(X))
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
5) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, activate^#(n__from(X)) -> c_12(from^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {activate^#(n__from(X)) -> c_12(from^#(X))}
Weak Rules:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__from(X)) -> c_12(from^#(X))}
and weakly orienting the rules
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__from(X)) -> c_12(from^#(X))}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [0]
fst^#(x1, x2) = [1] x1 + [1] x2 + [9]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [7]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [1] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15() = [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:
{ activate^#(n__from(X)) -> c_12(from^#(X))
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
6) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, add^#(0(), 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:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {add^#(0(), X) -> c_3()}
Weak Rules:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{add^#(0(), X) -> c_3()}
and weakly orienting the rules
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{add^#(0(), X) -> c_3()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [8]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [7]
c_15() = [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:
{ add^#(0(), X) -> c_3()
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
7) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, activate^#(X) -> c_15()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {activate^#(X) -> c_15()}
Weak Rules:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_15()}
and weakly orienting the rules
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_15()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [2]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [15]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [3]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [8]
c_14(x1) = [1] x1 + [3]
c_15() = [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:
{ activate^#(X) -> c_15()
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
8) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, fst^#(X1, X2) -> c_7()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {fst^#(X1, X2) -> c_7()}
Weak Rules:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{fst^#(X1, X2) -> c_7()}
and weakly orienting the rules
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{fst^#(X1, X2) -> c_7()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [4]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [7]
c_15() = [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:
{ fst^#(X1, X2) -> c_7()
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
9) { fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, add^#(X1, X2) -> c_9()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {add^#(X1, X2) -> c_9()}
Weak Rules:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{add^#(X1, X2) -> c_9()}
and weakly orienting the rules
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{add^#(X1, X2) -> c_9()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [4]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [7]
c_15() = [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:
{ add^#(X1, X2) -> c_9()
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
10)
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, len^#(nil()) -> c_5()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {len^#(nil()) -> c_5()}
Weak Rules:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{len^#(nil()) -> c_5()}
and weakly orienting the rules
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{len^#(nil()) -> c_5()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [8]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [1]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [1]
c_14(x1) = [1] x1 + [7]
c_15() = [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:
{ len^#(nil()) -> c_5()
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules
11)
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))
, len^#(X) -> c_10()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__fst(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [0] x1 + [0] x2 + [0]
len(x1) = [0] x1 + [0]
n__len(x1) = [0] x1 + [0]
fst^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
len^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15() = [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: {len^#(X) -> c_10()}
Weak Rules:
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{len^#(X) -> c_10()}
and weakly orienting the rules
{ fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{len^#(X) -> c_10()}
Details:
Interpretation Functions:
fst(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__fst(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
n__add(x1, x2) = [1] x1 + [1] x2 + [15]
len(x1) = [0] x1 + [0]
n__len(x1) = [1] x1 + [8]
fst^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
len^#(x1) = [1] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [0] x1 + [0]
c_13(x1) = [1] x1 + [11]
c_14(x1) = [1] x1 + [0]
c_15() = [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:
{ len^#(X) -> c_10()
, fst^#(s(X), cons(Y, Z)) -> c_1(activate^#(X), activate^#(Z))
, activate^#(n__fst(X1, X2)) -> c_11(fst^#(X1, X2))
, len^#(cons(X, Z)) -> c_6(activate^#(Z))
, activate^#(n__len(X)) -> c_14(len^#(X))
, add^#(s(X), Y) -> c_4(activate^#(X))
, activate^#(n__add(X1, X2)) -> c_13(add^#(X1, X2))}
Details:
The given problem does not contain any strict rules