Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ primes() -> sieve(from(s(s(0()))))
, from(X) -> cons(X, from(s(X)))
, head(cons(X, Y)) -> X
, tail(cons(X, Y)) -> Y
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y
, filter(s(s(X)), cons(Y, Z)) ->
if(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y))))
, sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: primes^#() -> c_0(sieve^#(from(s(s(0())))))
, 2: from^#(X) -> c_1(from^#(s(X)))
, 3: head^#(cons(X, Y)) -> c_2()
, 4: tail^#(cons(X, Y)) -> c_3()
, 5: if^#(true(), X, Y) -> c_4()
, 6: if^#(false(), X, Y) -> c_5()
, 7: filter^#(s(s(X)), cons(Y, Z)) ->
c_6(if^#(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y)))))
, 8: sieve^#(cons(X, Y)) -> c_7(filter^#(X, sieve(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8}->{7}.
* Path {1}->{8}: inherited
------------------------
This path is subsumed by the proof of path {1}->{8}->{7}.
* Path {1}->{8}->{7}: NA
----------------------
The usable rules for this path are:
{ from(X) -> cons(X, from(s(X)))
, sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
, filter(s(s(X)), cons(Y, Z)) ->
if(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y))))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {1}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_1(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {head^#(cons(X, Y)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
head^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(X, Y)) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
tail^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(true(), X, Y) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
[2 2] [0 0] [0 0] [7]
c_4() = [0]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(false(), X, Y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
[2 2] [0 0] [0 0] [7]
c_5() = [0]
[1]
2) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: primes^#() -> c_0(sieve^#(from(s(s(0())))))
, 2: from^#(X) -> c_1(from^#(s(X)))
, 3: head^#(cons(X, Y)) -> c_2()
, 4: tail^#(cons(X, Y)) -> c_3()
, 5: if^#(true(), X, Y) -> c_4()
, 6: if^#(false(), X, Y) -> c_5()
, 7: filter^#(s(s(X)), cons(Y, Z)) ->
c_6(if^#(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y)))))
, 8: sieve^#(cons(X, Y)) -> c_7(filter^#(X, sieve(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8}->{7}.
* Path {1}->{8}: inherited
------------------------
This path is subsumed by the proof of path {1}->{8}->{7}.
* Path {1}->{8}->{7}: NA
----------------------
The usable rules for this path are:
{ from(X) -> cons(X, from(s(X)))
, sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
, filter(s(s(X)), cons(Y, Z)) ->
if(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y))))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {1}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [3] x1 + [0]
c_1(x1) = [1] x1 + [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4() = [0]
c_5() = [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_1(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4() = [0]
c_5() = [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {head^#(cons(X, Y)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_2() = [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4() = [0]
c_5() = [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(X, Y)) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4() = [0]
c_5() = [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(true(), X, Y) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [7]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
c_4() = [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4() = [0]
c_5() = [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {if^#(false(), X, Y) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
c_5() = [1]
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ primes() -> sieve(from(s(s(0()))))
, from(X) -> cons(X, from(s(X)))
, head(cons(X, Y)) -> X
, tail(cons(X, Y)) -> Y
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y
, filter(s(s(X)), cons(Y, Z)) ->
if(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y))))
, sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: primes^#() -> c_0(sieve^#(from(s(s(0())))))
, 2: from^#(X) -> c_1(X, from^#(s(X)))
, 3: head^#(cons(X, Y)) -> c_2(X)
, 4: tail^#(cons(X, Y)) -> c_3(Y)
, 5: if^#(true(), X, Y) -> c_4(X)
, 6: if^#(false(), X, Y) -> c_5(Y)
, 7: filter^#(s(s(X)), cons(Y, Z)) ->
c_6(if^#(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y)))))
, 8: sieve^#(cons(X, Y)) -> c_7(X, filter^#(X, sieve(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^2)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8}->{7}.
* Path {1}->{8}: inherited
------------------------
This path is subsumed by the proof of path {1}->{8}->{7}.
* Path {1}->{8}->{7}: NA
----------------------
The usable rules for this path are:
{ from(X) -> cons(X, from(s(X)))
, sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
, filter(s(s(X)), cons(Y, Z)) ->
if(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y))))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {2}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_1(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_1(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(cons(X, Y)) -> c_2(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
head^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_2(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(X, Y)) -> c_3(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
tail^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_3(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [1 1] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(true(), X, Y) -> c_4(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
if^#(x1, x2, x3) = [2 2] x1 + [7 7] x2 + [0 0] x3 + [7]
[2 2] [7 7] [0 0] [3]
c_4(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
[0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
primes^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [1 1] x1 + [0]
[0 0] [0]
filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(false(), X, Y) -> c_5(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
if^#(x1, x2, x3) = [2 2] x1 + [0 0] x2 + [7 7] x3 + [7]
[2 2] [0 0] [7 7] [3]
c_5(x1) = [1 3] x1 + [0]
[1 1] [1]
2) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: primes^#() -> c_0(sieve^#(from(s(s(0())))))
, 2: from^#(X) -> c_1(X, from^#(s(X)))
, 3: head^#(cons(X, Y)) -> c_2(X)
, 4: tail^#(cons(X, Y)) -> c_3(Y)
, 5: if^#(true(), X, Y) -> c_4(X)
, 6: if^#(false(), X, Y) -> c_5(Y)
, 7: filter^#(s(s(X)), cons(Y, Z)) ->
c_6(if^#(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y)))))
, 8: sieve^#(cons(X, Y)) -> c_7(X, filter^#(X, sieve(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
->{3} [ YES(?,O(n^1)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8}->{7}.
* Path {1}->{8}: inherited
------------------------
This path is subsumed by the proof of path {1}->{8}->{7}.
* Path {1}->{8}->{7}: NA
----------------------
The usable rules for this path are:
{ from(X) -> cons(X, from(s(X)))
, sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
, filter(s(s(X)), cons(Y, Z)) ->
if(divides(s(s(X)), Y),
filter(s(s(X)), Z),
cons(Y, filter(X, sieve(Y))))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {2}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [3] x1 + [0]
c_1(x1, x2) = [2] x1 + [1] x2 + [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_1(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
head^#(x1) = [3] x1 + [0]
c_2(x1) = [1] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(cons(X, Y)) -> c_2(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_2(x1) = [1] x1 + [1]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [3] x1 + [0]
c_3(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(X, Y)) -> c_3(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [1]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(true(), X, Y) -> c_4(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [5]
if^#(x1, x2, x3) = [3] x1 + [7] x2 + [0] x3 + [0]
c_4(x1) = [1] x1 + [0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
primes() = [0]
sieve(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
false() = [0]
filter(x1, x2) = [0] x1 + [0] x2 + [0]
divides(x1, x2) = [0] x1 + [0] x2 + [0]
primes^#() = [0]
c_0(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(false(), X, Y) -> c_5(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [5]
if^#(x1, x2, x3) = [3] x1 + [0] x2 + [7] x3 + [0]
c_5(x1) = [1] x1 + [0]
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.