Tool CaT
stdout:
MAYBE
Problem:
cons(x,cons(y,z)) -> big()
inf(x) -> cons(x,inf(s(x)))
g(x,x) -> b()
Proof:
OpenTool IRC1
stdout:
MAYBE
Warning when parsing problem:
Unsupported strategy 'OUTERMOST'Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ cons(x, cons(y, z)) -> big()
, inf(x) -> cons(x, inf(s(x)))
, g(x, x) -> b()}
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: cons^#(x, cons(y, z)) -> c_0()
, 2: inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, 3: g^#(x, x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, cons^#(x, cons(y, z)) -> c_0()
, inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
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(cons) = {}, Uargs(inf) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(cons^#) = {}, Uargs(inf^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
big() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(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]
b() = [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]
c_0() = [0]
[0]
[0]
inf^#(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]
g^#(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]
c_2() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(x, x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [7]
[0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [7]
c_2() = [0]
[3]
[3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cons^#(x, cons(y, z)) -> c_0()
, 2: inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, 3: g^#(x, x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, cons^#(x, cons(y, z)) -> c_0()
, inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
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(cons) = {}, Uargs(inf) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(cons^#) = {}, Uargs(inf^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
big() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
b() = [0]
[0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [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: {g^#(x, x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [7]
[0 0] [0 0] [7]
c_2() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cons^#(x, cons(y, z)) -> c_0()
, 2: inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, 3: g^#(x, x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, cons^#(x, cons(y, z)) -> c_0()
, inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
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(cons) = {}, Uargs(inf) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(cons^#) = {}, Uargs(inf^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [0]
big() = [0]
inf(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
b() = [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
inf^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [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: {g^#(x, x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0] x1 + [0] x2 + [7]
c_2() = [0]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
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
Warning when parsing problem:
Unsupported strategy 'OUTERMOST'Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ cons(x, cons(y, z)) -> big()
, inf(x) -> cons(x, inf(s(x)))
, g(x, x) -> b()}
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: cons^#(x, cons(y, z)) -> c_0()
, 2: inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, 3: g^#(x, x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, cons^#(x, cons(y, z)) -> c_0()
, inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
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(cons) = {}, Uargs(inf) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(cons^#) = {}, Uargs(inf^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
big() = [0]
[0]
[0]
inf(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(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]
b() = [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]
c_0() = [0]
[0]
[0]
inf^#(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]
g^#(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]
c_2() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(x, x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [7]
[0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [7]
c_2() = [0]
[3]
[3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cons^#(x, cons(y, z)) -> c_0()
, 2: inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, 3: g^#(x, x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, cons^#(x, cons(y, z)) -> c_0()
, inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
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(cons) = {}, Uargs(inf) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(cons^#) = {}, Uargs(inf^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
big() = [0]
[0]
inf(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
b() = [0]
[0]
cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
inf^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [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: {g^#(x, x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [7]
[0 0] [0 0] [7]
c_2() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cons^#(x, cons(y, z)) -> c_0()
, 2: inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, 3: g^#(x, x) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ inf^#(x) -> c_1(cons^#(x, inf(s(x))))
, cons^#(x, cons(y, z)) -> c_0()
, inf(x) -> cons(x, inf(s(x)))
, cons(x, cons(y, z)) -> big()}
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(cons) = {}, Uargs(inf) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(cons^#) = {}, Uargs(inf^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [0]
big() = [0]
inf(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
b() = [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
inf^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [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: {g^#(x, x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0] x1 + [0] x2 + [7]
c_2() = [0]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
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.