Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.16 |
|---|
stdout:
MAYBE
Problem:
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
plus(x,0()) -> x
plus(0(),x) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(s(x),y) -> s(plus(x,y))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.16 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.16 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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: times^#(x, 0()) -> c_0()
, 2: times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, 3: plus^#(x, 0()) -> c_2()
, 4: plus^#(0(), x) -> c_3()
, 5: plus^#(x, s(y)) -> c_4(plus^#(x, y))
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ inherited ]
|
|->{3} [ MAYBE ]
|
|->{4} [ NA ]
|
`->{6,5} [ inherited ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(times) = {}, Uargs(s) = {}, Uargs(plus) = {},
Uargs(times^#) = {}, Uargs(c_1) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(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]
plus(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]
times^#(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]
plus^#(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() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(x, 0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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:
{ times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, plus^#(x, 0()) -> c_2()
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{4}: NA
-----------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}: inherited
--------------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{6,5}->{3}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}->{4}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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: times^#(x, 0()) -> c_0()
, 2: times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, 3: plus^#(x, 0()) -> c_2()
, 4: plus^#(0(), x) -> c_3()
, 5: plus^#(x, s(y)) -> c_4(plus^#(x, y))
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ inherited ]
|
|->{3} [ MAYBE ]
|
|->{4} [ NA ]
|
`->{6,5} [ inherited ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(times) = {}, Uargs(s) = {}, Uargs(plus) = {},
Uargs(times^#) = {}, Uargs(c_1) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(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]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times^#(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]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 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: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(x, 0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_0() = [0]
[1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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:
{ times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, plus^#(x, 0()) -> c_2()
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{4}: NA
-----------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}: inherited
--------------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{6,5}->{3}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}->{4}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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: times^#(x, 0()) -> c_0()
, 2: times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, 3: plus^#(x, 0()) -> c_2()
, 4: plus^#(0(), x) -> c_3()
, 5: plus^#(x, s(y)) -> c_4(plus^#(x, y))
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ inherited ]
|
|->{3} [ NA ]
|
|->{4} [ MAYBE ]
|
`->{6,5} [ inherited ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(times) = {}, Uargs(s) = {}, Uargs(plus) = {},
Uargs(times^#) = {}, Uargs(c_1) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(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: {times^#(x, 0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_0() = [1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{3}: NA
-----------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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:
{ times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, plus^#(0(), x) -> c_3()
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{6,5}: inherited
--------------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{6,5}->{3}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}->{4}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.16 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.16 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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: times^#(x, 0()) -> c_0()
, 2: times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, 3: plus^#(x, 0()) -> c_2(x)
, 4: plus^#(0(), x) -> c_3(x)
, 5: plus^#(x, s(y)) -> c_4(plus^#(x, y))
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ inherited ]
|
|->{3} [ MAYBE ]
|
|->{4} [ NA ]
|
`->{6,5} [ inherited ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(times) = {}, Uargs(s) = {}, Uargs(plus) = {},
Uargs(times^#) = {}, Uargs(c_1) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(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]
plus(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]
times^#(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]
plus^#(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(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]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(x, 0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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:
{ times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, plus^#(x, 0()) -> c_2(x)
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{4}: NA
-----------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}: inherited
--------------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{6,5}->{3}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}->{4}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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: times^#(x, 0()) -> c_0()
, 2: times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, 3: plus^#(x, 0()) -> c_2(x)
, 4: plus^#(0(), x) -> c_3(x)
, 5: plus^#(x, s(y)) -> c_4(plus^#(x, y))
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ inherited ]
|
|->{3} [ MAYBE ]
|
|->{4} [ NA ]
|
`->{6,5} [ inherited ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(times) = {}, Uargs(s) = {}, Uargs(plus) = {},
Uargs(times^#) = {}, Uargs(c_1) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(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]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times^#(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]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 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: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(x, 0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_0() = [0]
[1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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:
{ times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, plus^#(x, 0()) -> c_2(x)
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{4}: NA
-----------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}: inherited
--------------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{6,5}->{3}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}->{4}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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: times^#(x, 0()) -> c_0()
, 2: times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, 3: plus^#(x, 0()) -> c_2(x)
, 4: plus^#(0(), x) -> c_3(x)
, 5: plus^#(x, s(y)) -> c_4(plus^#(x, y))
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ inherited ]
|
|->{3} [ NA ]
|
|->{4} [ MAYBE ]
|
`->{6,5} [ inherited ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(times) = {}, Uargs(s) = {}, Uargs(plus) = {},
Uargs(times^#) = {}, Uargs(c_1) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(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: {times^#(x, 0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_0() = [1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{3}: NA
-----------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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:
{ times^#(x, s(y)) -> c_1(plus^#(times(x, y), x))
, plus^#(0(), x) -> c_3(x)
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{6,5}: inherited
--------------------------
This path is subsumed by the proof of path {2}->{6,5}->{4}.
* Path {2}->{6,5}->{3}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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.
* Path {2}->{6,5}->{4}: NA
------------------------
The usable rules for this path are:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.16 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.16 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.16 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.16 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.16 |
|---|
stdout:
MAYBE
We consider the following Problem:
Strict Trs:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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 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) '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) 'dp' failed due to the following reason:
We have computed the following dependency pairs
Strict Dependency Pairs:
{ times^#(x, 0()) -> c_1()
, times^#(x, s(y)) -> c_2(plus^#(times(x, y), x))
, plus^#(x, 0()) -> c_3(x)
, plus^#(0(), x) -> c_4(x)
, plus^#(x, s(y)) -> c_5(plus^#(x, y))
, plus^#(s(x), y) -> c_6(plus^#(x, y))}
We consider the following Problem:
Strict DPs:
{ times^#(x, 0()) -> c_1()
, times^#(x, s(y)) -> c_2(plus^#(times(x, y), x))
, plus^#(x, 0()) -> c_3(x)
, plus^#(0(), x) -> c_4(x)
, plus^#(x, s(y)) -> c_5(plus^#(x, y))
, plus^#(s(x), y) -> c_6(plus^#(x, y))}
Strict Trs:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(x, y))}
StartTerms: basic terms
Strategy: none
Certificate: MAYBE
Application of 'usablerules':
-----------------------------
All rules are usable.
No subproblems were generated.
Arrrr..Tool tup3irc
| Execution Time | 66.02863ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.16 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(x, 0()) -> x
, plus(0(), x) -> x
, plus(x, s(y)) -> s(plus(x, y))
, plus(s(x), y) -> s(plus(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..