Tool CaT
stdout:
MAYBE
Problem:
add(0(),x) -> x
add(s(x),y) -> s(add(x,y))
mult(0(),x) -> 0()
mult(s(x),y) -> add(y,mult(x,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:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, 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: add^#(0(), x) -> c_0()
, 2: add^#(s(x), y) -> c_1(add^#(x, y))
, 3: mult^#(0(), x) -> c_2()
, 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{3} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(mult^#) = {},
Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(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 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
mult(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]
add^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
mult^#(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]
c_3(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: innermost DP runtime-complexity with respect to
Strict Rules: {mult^#(0(), x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(mult^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
mult^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_2() = [0]
[1]
[1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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:
{ mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
, add^#(0(), x) -> c_0()
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}->{2}: inherited
------------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: add^#(0(), x) -> c_0()
, 2: add^#(s(x), y) -> c_1(add^#(x, y))
, 3: mult^#(0(), x) -> c_2()
, 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{3} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(mult^#) = {},
Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
mult(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
mult^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(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: {mult^#(0(), x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(mult^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
mult^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_2() = [0]
[1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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:
{ mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
, add^#(0(), x) -> c_0()
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}->{2}: inherited
------------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: add^#(0(), x) -> c_0()
, 2: add^#(s(x), y) -> c_1(add^#(x, y))
, 3: mult^#(0(), x) -> c_2()
, 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{3} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(mult^#) = {},
Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
mult(x1, x2) = [0] x1 + [0] x2 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
mult^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(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: {mult^#(0(), x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(mult^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
mult^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_2() = [1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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:
{ mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
, add^#(0(), x) -> c_0()
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}->{2}: inherited
------------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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.
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:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, 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: add^#(0(), x) -> c_0(x)
, 2: add^#(s(x), y) -> c_1(add^#(x, y))
, 3: mult^#(0(), x) -> c_2()
, 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{3} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
Uargs(add^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(mult^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(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 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
mult(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]
add^#(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(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]
mult^#(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]
c_3(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: {mult^#(0(), x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(mult^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
mult^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_2() = [0]
[1]
[1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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:
{ mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
, add^#(0(), x) -> c_0(x)
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}->{2}: inherited
------------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: add^#(0(), x) -> c_0(x)
, 2: add^#(s(x), y) -> c_1(add^#(x, y))
, 3: mult^#(0(), x) -> c_2()
, 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{3} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
Uargs(add^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(mult^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
mult(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
mult^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(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: {mult^#(0(), x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(mult^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
mult^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_2() = [0]
[1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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:
{ mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
, add^#(0(), x) -> c_0(x)
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}->{2}: inherited
------------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: add^#(0(), x) -> c_0(x)
, 2: add^#(s(x), y) -> c_1(add^#(x, y))
, 3: mult^#(0(), x) -> c_2()
, 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{3} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
Uargs(add^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(mult^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
mult(x1, x2) = [0] x1 + [0] x2 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
mult^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(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: {mult^#(0(), x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(mult^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
mult^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_2() = [1]
* Path {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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:
{ mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
, add^#(0(), x) -> c_0(x)
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}->{2}: inherited
------------------------
This path is subsumed by the proof of path {4}->{2}->{1}.
* Path {4}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, 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.
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 pair1rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
stdout:
MAYBE
We consider the following Problem:
Strict Trs:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: MAYBE
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Sequentially' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'matrix-interpretation of dimension 4 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
2) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
3) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Bounds with minimal-enrichment and initial automaton 'match' (timeout of 100.0 seconds)' failed due to the following reason:
match-boundness of the problem could not be verified.
2) 'Bounds with perSymbol-enrichment and initial automaton 'match' (timeout of 5.0 seconds)' failed due to the following reason:
match-boundness of the problem could not be verified.
2) 'dp' failed due to the following reason:
We have computed the following dependency pairs
Strict Dependency Pairs:
{ add^#(0(), x) -> c_1(x)
, add^#(s(x), y) -> c_2(add^#(x, y))
, mult^#(0(), x) -> c_3()
, mult^#(s(x), y) -> c_4(add^#(y, mult(x, y)))}
We consider the following Problem:
Strict DPs:
{ add^#(0(), x) -> c_1(x)
, add^#(s(x), y) -> c_2(add^#(x, y))
, mult^#(0(), x) -> c_3()
, mult^#(s(x), y) -> c_4(add^#(y, mult(x, y)))}
Strict Trs:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: MAYBE
Application of 'usablerules':
-----------------------------
All rules are usable.
No subproblems were generated.
Arrrr..Tool tup3irc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, mult(0(), x) -> 0()
, mult(s(x), y) -> add(y, mult(x, y))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..