Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.23 |
|---|
stdout:
MAYBE
Problem:
fac(0()) -> 1()
fac(s(x)) -> *(s(x),fac(x))
floop(0(),y) -> y
floop(s(x),y) -> floop(x,*(s(x),y))
*(x,0()) -> 0()
*(x,s(y)) -> +(*(x,y),x)
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
1() -> s(0())
fac(0()) -> s(0())
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.23 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.23 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, floop(0(), y) -> y
, floop(s(x), y) -> floop(x, *(s(x), y))
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, 1() -> s(0())
, fac(0()) -> s(0())}
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: fac^#(0()) -> c_0(1^#())
, 2: fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, 3: floop^#(0(), y) -> c_2()
, 4: floop^#(s(x), y) -> c_3(floop^#(x, *(s(x), y)))
, 5: *^#(x, 0()) -> c_4()
, 6: *^#(x, s(y)) -> c_5(+^#(*(x, y), x))
, 7: +^#(x, 0()) -> c_6()
, 8: +^#(x, s(y)) -> c_7(+^#(x, y))
, 9: 1^#() -> c_8()
, 10: fac^#(0()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
|->{5} [ MAYBE ]
|
`->{6} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
->{1} [ YES(?,O(1)) ]
|
`->{9} [ 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
1() = [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]
floop(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]
+(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]
fac^#(x1) = [0 0 0] x1 + [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]
1^#() = [0]
[0]
[0]
c_1(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]
floop^#(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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [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: {fac^#(0()) -> c_0(1^#())}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
fac^#(x1) = [0 2 2] x1 + [7]
[2 2 0] [7]
[0 0 0] [7]
c_0(x1) = [2 2 0] x1 + [1]
[0 0 0] [7]
[2 0 0] [3]
1^#() = [2]
[2]
[0]
* Path {1}->{9}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {}, Uargs(*^#) = {}, Uargs(floop^#) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
1() = [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]
floop(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]
+(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]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
1^#() = [0]
[0]
[0]
c_1(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]
floop^#(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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [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: {1^#() -> c_8()}
Weak Rules: {fac^#(0()) -> c_0(1^#())}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[6]
[2]
fac^#(x1) = [2 0 2] x1 + [7]
[0 2 0] [3]
[2 0 0] [7]
c_0(x1) = [2 2 0] x1 + [2]
[2 2 2] [3]
[2 2 0] [3]
1^#() = [2]
[2]
[2]
c_8() = [1]
[0]
[0]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, *^#(x, 0()) -> c_4()
, fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{6}: inherited
------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{7}: NA
----------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {2}->{6}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{3}.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), 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 {10}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
1() = [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]
floop(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]
+(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]
fac^#(x1) = [0 0 0] x1 + [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]
1^#() = [0]
[0]
[0]
c_1(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]
floop^#(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]
c_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [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: {fac^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
fac^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_9() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fac^#(0()) -> c_0(1^#())
, 2: fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, 3: floop^#(0(), y) -> c_2()
, 4: floop^#(s(x), y) -> c_3(floop^#(x, *(s(x), y)))
, 5: *^#(x, 0()) -> c_4()
, 6: *^#(x, s(y)) -> c_5(+^#(*(x, y), x))
, 7: +^#(x, 0()) -> c_6()
, 8: +^#(x, s(y)) -> c_7(+^#(x, y))
, 9: 1^#() -> c_8()
, 10: fac^#(0()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
|->{5} [ MAYBE ]
|
`->{6} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
->{1} [ YES(?,O(1)) ]
|
`->{9} [ 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
1() = [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]
floop(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
floop^#(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]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [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: {fac^#(0()) -> c_0(1^#())}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fac^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_0(x1) = [0 2] x1 + [3]
[0 2] [3]
1^#() = [0]
[2]
* Path {1}->{9}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {}, Uargs(*^#) = {}, Uargs(floop^#) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
1() = [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]
floop(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
1^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
floop^#(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]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [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: {1^#() -> c_8()}
Weak Rules: {fac^#(0()) -> c_0(1^#())}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fac^#(x1) = [0 0] x1 + [7]
[1 2] [7]
c_0(x1) = [2 0] x1 + [2]
[2 2] [3]
1^#() = [2]
[2]
c_8() = [1]
[0]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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:
{ fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, *^#(x, 0()) -> c_4()
, fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{6}: inherited
------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{7}: NA
----------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {2}->{6}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{3}.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), 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 {10}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
1() = [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]
floop(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
floop^#(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]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [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: {fac^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fac^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_9() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fac^#(0()) -> c_0(1^#())
, 2: fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, 3: floop^#(0(), y) -> c_2()
, 4: floop^#(s(x), y) -> c_3(floop^#(x, *(s(x), y)))
, 5: *^#(x, 0()) -> c_4()
, 6: *^#(x, s(y)) -> c_5(+^#(*(x, y), x))
, 7: +^#(x, 0()) -> c_6()
, 8: +^#(x, s(y)) -> c_7(+^#(x, y))
, 9: 1^#() -> c_8()
, 10: fac^#(0()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{3} [ MAYBE ]
->{2} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
->{1} [ YES(?,O(1)) ]
|
`->{9} [ 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
floop(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fac^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#() = [0]
c_1(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
floop^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [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: {fac^#(0()) -> c_0(1^#())}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
fac^#(x1) = [2] x1 + [5]
c_0(x1) = [2] x1 + [3]
1^#() = [2]
* Path {1}->{9}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {}, Uargs(*^#) = {}, Uargs(floop^#) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
floop(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fac^#(x1) = [0] x1 + [0]
c_0(x1) = [1] x1 + [0]
1^#() = [0]
c_1(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
floop^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [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: {1^#() -> c_8()}
Weak Rules: {fac^#(0()) -> c_0(1^#())}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
fac^#(x1) = [0] x1 + [6]
c_0(x1) = [2] x1 + [2]
1^#() = [2]
c_8() = [1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{5}: NA
-----------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {2}->{6}: inherited
------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{7}: NA
----------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {2}->{6}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{3}.
* Path {4}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), 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:
{ floop^#(s(x), y) -> c_3(floop^#(x, *(s(x), y)))
, floop^#(0(), y) -> c_2()
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {10}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
floop(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fac^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#() = [0]
c_1(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
floop^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [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: {fac^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
fac^#(x1) = [1] x1 + [7]
c_9() = [1]
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 2.23 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 2.23 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, floop(0(), y) -> y
, floop(s(x), y) -> floop(x, *(s(x), y))
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, 1() -> s(0())
, fac(0()) -> s(0())}
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: fac^#(0()) -> c_0(1^#())
, 2: fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, 3: floop^#(0(), y) -> c_2(y)
, 4: floop^#(s(x), y) -> c_3(floop^#(x, *(s(x), y)))
, 5: *^#(x, 0()) -> c_4()
, 6: *^#(x, s(y)) -> c_5(+^#(*(x, y), x))
, 7: +^#(x, 0()) -> c_6(x)
, 8: +^#(x, s(y)) -> c_7(+^#(x, y))
, 9: 1^#() -> c_8()
, 10: fac^#(0()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
|->{5} [ MAYBE ]
|
`->{6} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
->{1} [ YES(?,O(1)) ]
|
`->{9} [ 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
1() = [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]
floop(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]
+(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]
fac^#(x1) = [0 0 0] x1 + [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]
1^#() = [0]
[0]
[0]
c_1(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]
floop^#(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() = [0]
[0]
[0]
c_5(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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [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: {fac^#(0()) -> c_0(1^#())}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
fac^#(x1) = [0 2 2] x1 + [7]
[2 2 0] [7]
[0 0 0] [7]
c_0(x1) = [2 2 0] x1 + [1]
[0 0 0] [7]
[2 0 0] [3]
1^#() = [2]
[2]
[0]
* Path {1}->{9}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {}, Uargs(*^#) = {}, Uargs(floop^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
1() = [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]
floop(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]
+(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]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
1^#() = [0]
[0]
[0]
c_1(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]
floop^#(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() = [0]
[0]
[0]
c_5(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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [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: {1^#() -> c_8()}
Weak Rules: {fac^#(0()) -> c_0(1^#())}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[6]
[2]
fac^#(x1) = [2 0 2] x1 + [7]
[0 2 0] [3]
[2 0 0] [7]
c_0(x1) = [2 2 0] x1 + [2]
[2 2 2] [3]
[2 2 0] [3]
1^#() = [2]
[2]
[2]
c_8() = [1]
[0]
[0]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, *^#(x, 0()) -> c_4()
, fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{6}: inherited
------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{7}: NA
----------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {2}->{6}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{3}.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), 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 {10}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
1() = [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]
floop(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]
+(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]
fac^#(x1) = [0 0 0] x1 + [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]
1^#() = [0]
[0]
[0]
c_1(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]
floop^#(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() = [0]
[0]
[0]
c_5(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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [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: {fac^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
fac^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_9() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fac^#(0()) -> c_0(1^#())
, 2: fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, 3: floop^#(0(), y) -> c_2(y)
, 4: floop^#(s(x), y) -> c_3(floop^#(x, *(s(x), y)))
, 5: *^#(x, 0()) -> c_4()
, 6: *^#(x, s(y)) -> c_5(+^#(*(x, y), x))
, 7: +^#(x, 0()) -> c_6(x)
, 8: +^#(x, s(y)) -> c_7(+^#(x, y))
, 9: 1^#() -> c_8()
, 10: fac^#(0()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
|->{5} [ MAYBE ]
|
`->{6} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
->{1} [ YES(?,O(1)) ]
|
`->{9} [ 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
1() = [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]
floop(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
floop^#(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() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [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: {fac^#(0()) -> c_0(1^#())}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fac^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_0(x1) = [0 2] x1 + [3]
[0 2] [3]
1^#() = [0]
[2]
* Path {1}->{9}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {}, Uargs(*^#) = {}, Uargs(floop^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
1() = [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]
floop(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
1^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
floop^#(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() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [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: {1^#() -> c_8()}
Weak Rules: {fac^#(0()) -> c_0(1^#())}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fac^#(x1) = [0 0] x1 + [7]
[1 2] [7]
c_0(x1) = [2 0] x1 + [2]
[2 2] [3]
1^#() = [2]
[2]
c_8() = [1]
[0]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{5}: MAYBE
--------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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:
{ fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, *^#(x, 0()) -> c_4()
, fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {2}->{6}: inherited
------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{7}: NA
----------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {2}->{6}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{3}.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), 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 {10}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
1() = [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]
floop(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
+(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fac^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
*^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
floop^#(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() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
+^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [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: {fac^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
fac^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_9() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fac^#(0()) -> c_0(1^#())
, 2: fac^#(s(x)) -> c_1(*^#(s(x), fac(x)))
, 3: floop^#(0(), y) -> c_2(y)
, 4: floop^#(s(x), y) -> c_3(floop^#(x, *(s(x), y)))
, 5: *^#(x, 0()) -> c_4()
, 6: *^#(x, s(y)) -> c_5(+^#(*(x, y), x))
, 7: +^#(x, 0()) -> c_6(x)
, 8: +^#(x, s(y)) -> c_7(+^#(x, y))
, 9: 1^#() -> c_8()
, 10: fac^#(0()) -> c_9()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(1)) ]
->{4} [ inherited ]
|
`->{3} [ MAYBE ]
->{2} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
->{1} [ YES(?,O(1)) ]
|
`->{9} [ 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
floop(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fac^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#() = [0]
c_1(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
floop^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [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: {fac^#(0()) -> c_0(1^#())}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
fac^#(x1) = [2] x1 + [5]
c_0(x1) = [2] x1 + [3]
1^#() = [2]
* Path {1}->{9}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {}, Uargs(*^#) = {}, Uargs(floop^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
floop(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fac^#(x1) = [0] x1 + [0]
c_0(x1) = [1] x1 + [0]
1^#() = [0]
c_1(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
floop^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [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: {1^#() -> c_8()}
Weak Rules: {fac^#(0()) -> c_0(1^#())}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}, Uargs(c_0) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
fac^#(x1) = [0] x1 + [6]
c_0(x1) = [2] x1 + [2]
1^#() = [2]
c_8() = [1]
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{5}: NA
-----------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {2}->{6}: inherited
------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{7}: NA
----------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {2}->{6}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {2}->{6}->{8}->{7}.
* Path {2}->{6}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, fac(0()) -> s(0())
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, 1() -> s(0())
, +(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 {4}: inherited
-------------------
This path is subsumed by the proof of path {4}->{3}.
* Path {4}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), 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:
{ floop^#(s(x), y) -> c_3(floop^#(x, *(s(x), y)))
, floop^#(0(), y) -> c_2(y)
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {10}: 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(fac) = {}, Uargs(s) = {}, Uargs(*) = {}, Uargs(floop) = {},
Uargs(+) = {}, Uargs(fac^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(*^#) = {}, Uargs(floop^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_5) = {}, Uargs(+^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fac(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
floop(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
fac^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
1^#() = [0]
c_1(x1) = [0] x1 + [0]
*^#(x1, x2) = [0] x1 + [0] x2 + [0]
floop^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [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: {fac^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(fac^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
fac^#(x1) = [1] x1 + [7]
c_9() = [1]
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 2.23 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, floop(0(), y) -> y
, floop(s(x), y) -> floop(x, *(s(x), y))
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, 1() -> s(0())
, fac(0()) -> s(0())}
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 2.23 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, floop(0(), y) -> y
, floop(s(x), y) -> floop(x, *(s(x), y))
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, 1() -> s(0())
, fac(0()) -> s(0())}
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 2.23 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, floop(0(), y) -> y
, floop(s(x), y) -> floop(x, *(s(x), y))
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, 1() -> s(0())
, fac(0()) -> s(0())}
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 2.23 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, floop(0(), y) -> y
, floop(s(x), y) -> floop(x, *(s(x), y))
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, 1() -> s(0())
, fac(0()) -> s(0())}
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 | TIMEOUT |
|---|
| Input | SK90 2.23 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, floop(0(), y) -> y
, floop(s(x), y) -> floop(x, *(s(x), y))
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, 1() -> s(0())
, fac(0()) -> s(0())}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
| Execution Time | 60.066135ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 2.23 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ fac(0()) -> 1()
, fac(s(x)) -> *(s(x), fac(x))
, floop(0(), y) -> y
, floop(s(x), y) -> floop(x, *(s(x), y))
, *(x, 0()) -> 0()
, *(x, s(y)) -> +(*(x, y), x)
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, 1() -> s(0())
, fac(0()) -> s(0())}
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..