Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.25 |
|---|
stdout:
MAYBE
Problem:
fib(0()) -> 0()
fib(s(0())) -> s(0())
fib(s(s(x))) -> +(fib(s(x)),fib(x))
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.25 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.25 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, 4: +^#(x, 0()) -> c_3()
, 5: +^#(x, s(y)) -> c_4(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{4} [ NA ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{2} [ YES(?,O(1)) ]
->{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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [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]
+(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]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(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(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: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
fib^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: 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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [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]
+(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]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(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(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: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fib^#(x1) = [2 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [7]
c_1() = [0]
[1]
[1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{4}: NA
-----------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 {3}->{5}: inherited
------------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, 4: +^#(x, 0()) -> c_3()
, 5: +^#(x, s(y)) -> c_4(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{2} [ YES(?,O(1)) ]
->{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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fib^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: 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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [0 1] x1 + [2]
[0 0] [2]
fib^#(x1) = [2 2] x1 + [3]
[2 2] [3]
c_1() = [0]
[1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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:
{ fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, +^#(x, 0()) -> c_3()
, fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {3}->{5}: inherited
------------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, 4: +^#(x, 0()) -> c_3()
, 5: +^#(x, s(y)) -> c_4(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{2} [ YES(?,O(1)) ]
->{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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
fib^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: 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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
fib^#(x1) = [2] x1 + [7]
c_1() = [0]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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:
{ fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, +^#(x, 0()) -> c_3()
, fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {3}->{5}: inherited
------------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 | SK90 2.25 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.25 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, 4: +^#(x, 0()) -> c_3(x)
, 5: +^#(x, s(y)) -> c_4(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{4} [ NA ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{2} [ YES(?,O(1)) ]
->{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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [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]
+(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]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(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(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]
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: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
fib^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: 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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [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]
+(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]
fib^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
+^#(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(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]
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: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fib^#(x1) = [2 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [7]
c_1() = [0]
[1]
[1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{4}: NA
-----------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 {3}->{5}: inherited
------------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, 4: +^#(x, 0()) -> c_3(x)
, 5: +^#(x, s(y)) -> c_4(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{2} [ YES(?,O(1)) ]
->{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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] 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: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fib^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: 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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] 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: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [0 1] x1 + [2]
[0 0] [2]
fib^#(x1) = [2 2] x1 + [3]
[2 2] [3]
c_1() = [0]
[1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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:
{ fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, +^#(x, 0()) -> c_3(x)
, fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {3}->{5}: inherited
------------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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: fib^#(0()) -> c_0()
, 2: fib^#(s(0())) -> c_1()
, 3: fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, 4: +^#(x, 0()) -> c_3(x)
, 5: +^#(x, s(y)) -> c_4(+^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{4} [ MAYBE ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{2} [ YES(?,O(1)) ]
->{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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] 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: {fib^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
fib^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: 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(fib) = {}, Uargs(s) = {}, Uargs(+) = {}, Uargs(fib^#) = {},
Uargs(c_2) = {}, Uargs(+^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] 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: {fib^#(s(0())) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(fib^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
fib^#(x1) = [2] x1 + [7]
c_1() = [0]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{4}: MAYBE
--------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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:
{ fib^#(s(s(x))) -> c_2(+^#(fib(s(x)), fib(x)))
, +^#(x, 0()) -> c_3(x)
, fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {3}->{5}: inherited
------------------------
This path is subsumed by the proof of path {3}->{5}->{4}.
* Path {3}->{5}->{4}: NA
----------------------
The usable rules for this path are:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 | SK90 2.25 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 | SK90 2.25 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 | SK90 2.25 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 | SK90 2.25 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 | SK90 2.25 |
|---|
stdout:
MAYBE
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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:
{ fib^#(0()) -> c_1()
, fib^#(s(0())) -> c_2()
, fib^#(s(s(x))) -> c_3(+^#(fib(s(x)), fib(x)))
, +^#(x, 0()) -> c_4(x)
, +^#(x, s(y)) -> c_5(+^#(x, y))}
We consider the following Problem:
Strict DPs:
{ fib^#(0()) -> c_1()
, fib^#(s(0())) -> c_2()
, fib^#(s(s(x))) -> c_3(+^#(fib(s(x)), fib(x)))
, +^#(x, 0()) -> c_4(x)
, +^#(x, s(y)) -> c_5(+^#(x, y))}
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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 | 60.072556ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 2.25 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fib(0()) -> 0()
, fib(s(0())) -> s(0())
, fib(s(s(x))) -> +(fib(s(x)), fib(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(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..