Tool CaT
stdout:
MAYBE
Problem:
f(x,c(x),c(y)) -> f(y,y,f(y,x,y))
f(s(x),y,z) -> f(x,s(c(y)),c(z))
f(c(x),x,y) -> c(y)
g(x,y) -> x
g(x,y) -> y
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)
, g(x, y) -> x
, g(x, y) -> 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: f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, 2: f^#(s(x), y, z) -> c_1(f^#(x, s(c(y)), c(z)))
, 3: f^#(c(x), x, y) -> c_2()
, 4: g^#(x, y) -> c_3()
, 5: g^#(x, y) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{2} [ inherited ]
| |
| `->{3} [ NA ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}->{3}: NA
----------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(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 {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
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:
{ f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, f^#(c(x), x, y) -> c_2()
, f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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(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]
f^#(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]
c_2() = [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_3() = [0]
[0]
[0]
c_4() = [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, y) -> c_3()}
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_3() = [0]
[3]
[3]
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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(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]
f^#(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]
c_2() = [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_3() = [0]
[0]
[0]
c_4() = [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, y) -> c_4()}
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_4() = [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: f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, 2: f^#(s(x), y, z) -> c_1(f^#(x, s(c(y)), c(z)))
, 3: f^#(c(x), x, y) -> c_2()
, 4: g^#(x, y) -> c_3()
, 5: g^#(x, y) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{2} [ inherited ]
| |
| `->{3} [ NA ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}->{3}: NA
----------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(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 {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
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:
{ f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, f^#(c(x), x, y) -> c_2()
, f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c(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]
f^#(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]
c_2() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [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, y) -> c_3()}
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_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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c(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]
f^#(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]
c_2() = [0]
[0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [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, y) -> c_4()}
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_4() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, 2: f^#(s(x), y, z) -> c_1(f^#(x, s(c(y)), c(z)))
, 3: f^#(c(x), x, y) -> c_2()
, 4: g^#(x, y) -> c_3()
, 5: g^#(x, y) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{2} [ inherited ]
| |
| `->{3} [ NA ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}->{3}: NA
----------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(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 {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
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:
{ f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, f^#(c(x), x, y) -> c_2()
, f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [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, y) -> c_3()}
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_3() = [0]
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [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, y) -> c_4()}
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_4() = [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
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)
, g(x, y) -> x
, g(x, y) -> 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: f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, 2: f^#(s(x), y, z) -> c_1(f^#(x, s(c(y)), c(z)))
, 3: f^#(c(x), x, y) -> c_2(y)
, 4: g^#(x, y) -> c_3(x)
, 5: g^#(x, y) -> c_4(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{2} [ inherited ]
| |
| `->{3} [ NA ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}->{3}: NA
----------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(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 {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
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:
{ f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, f^#(c(x), x, y) -> c_2(y)
, f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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(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]
f^#(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]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [1 1 1] 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]
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, y) -> c_3(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [7 7 7] x1 + [0 0 0] x2 + [7]
[7 7 7] [0 0 0] [7]
[7 7 7] [0 0 0] [7]
c_3(x1) = [3 3 3] x1 + [0]
[3 1 3] [1]
[1 1 1] [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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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(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]
f^#(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]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
c_4(x1) = [1 1 1] 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: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(x, y) -> c_4(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0 0] x1 + [7 7 7] x2 + [7]
[0 0 0] [7 7 7] [7]
[0 0 0] [7 7 7] [7]
c_4(x1) = [3 3 3] x1 + [0]
[3 1 3] [1]
[1 1 1] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, 2: f^#(s(x), y, z) -> c_1(f^#(x, s(c(y)), c(z)))
, 3: f^#(c(x), x, y) -> c_2(y)
, 4: g^#(x, y) -> c_3(x)
, 5: g^#(x, y) -> c_4(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{2} [ inherited ]
| |
| `->{3} [ NA ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}->{3}: NA
----------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(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 {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
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:
{ f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, f^#(c(x), x, y) -> c_2(y)
, f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c(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]
f^#(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]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [1 1] x1 + [0]
[0 0] [0]
c_4(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: DP runtime-complexity with respect to
Strict Rules: {g^#(x, y) -> c_3(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [7 7] x1 + [0 0] x2 + [7]
[7 7] [0 0] [7]
c_3(x1) = [1 3] x1 + [0]
[3 1] [3]
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c(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]
f^#(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]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 1] 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: DP runtime-complexity with respect to
Strict Rules: {g^#(x, y) -> c_4(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0 0] x1 + [7 7] x2 + [7]
[0 0] [7 7] [7]
c_4(x1) = [1 3] x1 + [0]
[3 1] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, 2: f^#(s(x), y, z) -> c_1(f^#(x, s(c(y)), c(z)))
, 3: f^#(c(x), x, y) -> c_2(y)
, 4: g^#(x, y) -> c_3(x)
, 5: g^#(x, y) -> c_4(y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{2} [ inherited ]
| |
| `->{3} [ NA ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}.
* Path {1}->{2}->{3}: NA
----------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(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 {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
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:
{ f^#(x, c(x), c(y)) -> c_0(f^#(y, y, f(y, x, y)))
, f^#(c(x), x, y) -> c_2(y)
, f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
, f(s(x), y, z) -> f(x, s(c(y)), c(z))
, f(c(x), x, y) -> c(y)}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_3(x1) = [1] x1 + [0]
c_4(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: DP runtime-complexity with respect to
Strict Rules: {g^#(x, y) -> c_3(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [7] x1 + [0] x2 + [7]
c_3(x1) = [1] x1 + [0]
* 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(f) = {}, Uargs(c) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [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, y) -> c_4(y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1, x2) = [0] x1 + [7] x2 + [7]
c_4(x1) = [1] x1 + [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.