'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ terms(N) -> cons(recip(sqr(N)))
, sqr(0()) -> 0()
, sqr(s()) -> s()
, dbl(0()) -> 0()
, dbl(s()) -> s()
, add(0(), X) -> X
, add(s(), Y) -> s()
, first(0(), X) -> nil()
, first(s(), cons(Y)) -> cons(Y)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ terms^#(N) -> c_0(sqr^#(N))
, sqr^#(0()) -> c_1()
, sqr^#(s()) -> c_2()
, dbl^#(0()) -> c_3()
, dbl^#(s()) -> c_4()
, add^#(0(), X) -> c_5()
, add^#(s(), Y) -> c_6()
, first^#(0(), X) -> c_7()
, first^#(s(), cons(Y)) -> c_8()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{terms^#(N) -> c_0(sqr^#(N))}
==> {sqr^#(s()) -> c_2()}
{terms^#(N) -> c_0(sqr^#(N))}
==> {sqr^#(0()) -> c_1()}
We consider the following path(s):
1) { terms^#(N) -> c_0(sqr^#(N))
, sqr^#(0()) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {sqr^#(0()) -> c_1()}
Weak Rules: {terms^#(N) -> c_0(sqr^#(N))}
Details:
We apply the weight gap principle, strictly orienting the rules
{sqr^#(0()) -> c_1()}
and weakly orienting the rules
{terms^#(N) -> c_0(sqr^#(N))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{sqr^#(0()) -> c_1()}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
sqr^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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:
{ sqr^#(0()) -> c_1()
, terms^#(N) -> c_0(sqr^#(N))}
Details:
The given problem does not contain any strict rules
2) { terms^#(N) -> c_0(sqr^#(N))
, sqr^#(s()) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {sqr^#(s()) -> c_2()}
Weak Rules: {terms^#(N) -> c_0(sqr^#(N))}
Details:
We apply the weight gap principle, strictly orienting the rules
{sqr^#(s()) -> c_2()}
and weakly orienting the rules
{terms^#(N) -> c_0(sqr^#(N))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{sqr^#(s()) -> c_2()}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
sqr^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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:
{ sqr^#(s()) -> c_2()
, terms^#(N) -> c_0(sqr^#(N))}
Details:
The given problem does not contain any strict rules
3) {terms^#(N) -> c_0(sqr^#(N))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {terms^#(N) -> c_0(sqr^#(N))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{terms^#(N) -> c_0(sqr^#(N))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{terms^#(N) -> c_0(sqr^#(N))}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
sqr^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {terms^#(N) -> c_0(sqr^#(N))}
Details:
The given problem does not contain any strict rules
4) {dbl^#(0()) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {dbl^#(0()) -> c_3()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{dbl^#(0()) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{dbl^#(0()) -> c_3()}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {dbl^#(0()) -> c_3()}
Details:
The given problem does not contain any strict rules
5) {add^#(0(), X) -> c_5()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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_5()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{add^#(0(), X) -> c_5()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{add^#(0(), X) -> c_5()}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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_5()}
Details:
The given problem does not contain any strict rules
6) {add^#(s(), Y) -> c_6()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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^#(s(), Y) -> c_6()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{add^#(s(), Y) -> c_6()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{add^#(s(), Y) -> c_6()}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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^#(s(), Y) -> c_6()}
Details:
The given problem does not contain any strict rules
7) {first^#(s(), cons(Y)) -> c_8()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {first^#(s(), cons(Y)) -> c_8()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{first^#(s(), cons(Y)) -> c_8()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{first^#(s(), cons(Y)) -> c_8()}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [1] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_7() = [0]
c_8() = [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: {first^#(s(), cons(Y)) -> c_8()}
Details:
The given problem does not contain any strict rules
8) {dbl^#(s()) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {dbl^#(s()) -> c_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{dbl^#(s()) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{dbl^#(s()) -> c_4()}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {dbl^#(s()) -> c_4()}
Details:
The given problem does not contain any strict rules
9) {first^#(0(), X) -> c_7()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [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: {first^#(0(), X) -> c_7()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{first^#(0(), X) -> c_7()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{first^#(0(), X) -> c_7()}
Details:
Interpretation Functions:
terms(x1) = [0] x1 + [0]
cons(x1) = [0] x1 + [0]
recip(x1) = [0] x1 + [0]
sqr(x1) = [0] x1 + [0]
0() = [0]
s() = [0]
dbl(x1) = [0] x1 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
first(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
terms^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sqr^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
dbl^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_7() = [0]
c_8() = [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: {first^#(0(), X) -> c_7()}
Details:
The given problem does not contain any strict rules