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:
{ filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
, filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
, sieve(cons(0(), Y)) -> cons(0(), sieve(Y))
, sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
, nats(N) -> cons(N, nats(s(N)))
, zprimes() -> sieve(nats(s(s(0()))))}
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: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
, 2: filter^#(cons(X, Y), s(N), M) -> c_1(filter^#(Y, N, M))
, 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
, 4: sieve^#(cons(s(N), Y)) -> c_3(sieve^#(filter(Y, N, N)))
, 5: nats^#(N) -> c_4(nats^#(s(N)))
, 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{3,4} [ NA ]
->{5} [ MAYBE ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {1}, Uargs(c_1) = {1}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
cons(x1, x2) = [1 3 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sieve(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nats(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zprimes() = [0]
[0]
[0]
filter^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sieve^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nats^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zprimes^#() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {1}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 0] [0]
sieve(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nats(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zprimes() = [0]
[0]
[0]
filter^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sieve^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nats^#(x1) = [3 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
zprimes^#() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {nats^#(N) -> c_4(nats^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3,4}.
* Path {6}->{3,4}: NA
-------------------
The usable rules for this path are:
{ nats(N) -> cons(N, nats(s(N)))
, filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
, filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}
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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
, 2: filter^#(cons(X, Y), s(N), M) -> c_1(filter^#(Y, N, M))
, 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
, 4: sieve^#(cons(s(N), Y)) -> c_3(sieve^#(filter(Y, N, N)))
, 5: nats^#(N) -> c_4(nats^#(s(N)))
, 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{3,4} [ NA ]
->{5} [ MAYBE ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {1}, Uargs(c_1) = {1}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
cons(x1, x2) = [1 3] x1 + [1 1] x2 + [0]
[0 1] [0 1] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
nats(x1) = [0 0] x1 + [0]
[0 0] [0]
zprimes() = [0]
[0]
filter^#(x1, x2, x3) = [1 3] x1 + [0 0] x2 + [1 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
nats^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
zprimes^#() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {1}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
nats(x1) = [0 0] x1 + [0]
[0 0] [0]
zprimes() = [0]
[0]
filter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
nats^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
zprimes^#() = [0]
[0]
c_5(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: {nats^#(N) -> c_4(nats^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3,4}.
* Path {6}->{3,4}: NA
-------------------
The usable rules for this path are:
{ nats(N) -> cons(N, nats(s(N)))
, filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
, filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}
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.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
, 2: filter^#(cons(X, Y), s(N), M) -> c_1(filter^#(Y, N, M))
, 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
, 4: sieve^#(cons(s(N), Y)) -> c_3(sieve^#(filter(Y, N, N)))
, 5: nats^#(N) -> c_4(nats^#(s(N)))
, 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{3,4} [ NA ]
->{5} [ MAYBE ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {1}, Uargs(c_1) = {1}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [3] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
nats^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {1}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
sieve^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
nats^#(x1) = [3] x1 + [0]
c_4(x1) = [1] x1 + [0]
zprimes^#() = [0]
c_5(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: {nats^#(N) -> c_4(nats^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3,4}.
* Path {6}->{3,4}: NA
-------------------
The usable rules for this path are:
{ nats(N) -> cons(N, nats(s(N)))
, filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
, filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}
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.
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) '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:
{ filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
, filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
, sieve(cons(0(), Y)) -> cons(0(), sieve(Y))
, sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
, nats(N) -> cons(N, nats(s(N)))
, zprimes() -> sieve(nats(s(s(0()))))}
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: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
, 2: filter^#(cons(X, Y), s(N), M) -> c_1(X, filter^#(Y, N, M))
, 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
, 4: sieve^#(cons(s(N), Y)) -> c_3(N, sieve^#(filter(Y, N, N)))
, 5: nats^#(N) -> c_4(N, nats^#(s(N)))
, 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{3,4} [ NA ]
->{5} [ MAYBE ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {1}, Uargs(c_1) = {2}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
cons(x1, x2) = [1 3 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
sieve(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nats(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zprimes() = [0]
[0]
[0]
filter^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
sieve^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nats^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
zprimes^#() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {2}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 1 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
sieve(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nats(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zprimes() = [0]
[0]
[0]
filter^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sieve^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
nats^#(x1) = [1 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_4(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
zprimes^#() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {nats^#(N) -> c_4(N, nats^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3,4}.
* Path {6}->{3,4}: NA
-------------------
The usable rules for this path are:
{ nats(N) -> cons(N, nats(s(N)))
, filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
, filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}
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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
, 2: filter^#(cons(X, Y), s(N), M) -> c_1(X, filter^#(Y, N, M))
, 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
, 4: sieve^#(cons(s(N), Y)) -> c_3(N, sieve^#(filter(Y, N, N)))
, 5: nats^#(N) -> c_4(N, nats^#(s(N)))
, 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{3,4} [ NA ]
->{5} [ MAYBE ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {1}, Uargs(c_1) = {2}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
[0 1] [0 1] [0]
0() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 1] [0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
nats(x1) = [0 0] x1 + [0]
[0 0] [0]
zprimes() = [0]
[0]
filter^#(x1, x2, x3) = [3 3] x1 + [3 3] x2 + [0 3] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nats^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
zprimes^#() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {2}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
sieve(x1) = [0 0] x1 + [0]
[0 0] [0]
nats(x1) = [0 0] x1 + [0]
[0 0] [0]
zprimes() = [0]
[0]
filter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sieve^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nats^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_4(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
zprimes^#() = [0]
[0]
c_5(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: DP runtime-complexity with respect to
Strict Rules: {nats^#(N) -> c_4(N, nats^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3,4}.
* Path {6}->{3,4}: NA
-------------------
The usable rules for this path are:
{ nats(N) -> cons(N, nats(s(N)))
, filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
, filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}
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.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
, 2: filter^#(cons(X, Y), s(N), M) -> c_1(X, filter^#(Y, N, M))
, 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
, 4: sieve^#(cons(s(N), Y)) -> c_3(N, sieve^#(filter(Y, N, N)))
, 5: nats^#(N) -> c_4(N, nats^#(s(N)))
, 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
`->{3,4} [ NA ]
->{5} [ MAYBE ]
->{1,2} [ NA ]
Sub-problems:
-------------
* Path {1,2}: NA
--------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {1}, Uargs(c_1) = {2}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [3] x1 + [0] x2 + [3] x3 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [0]
sieve^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
nats^#(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
Uargs(c_4) = {2}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
sieve^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
nats^#(x1) = [3] x1 + [0]
c_4(x1, x2) = [2] x1 + [1] x2 + [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [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: {nats^#(N) -> c_4(N, nats^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3,4}.
* Path {6}->{3,4}: NA
-------------------
The usable rules for this path are:
{ nats(N) -> cons(N, nats(s(N)))
, filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
, filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}
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.
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.