Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.59 |
|---|
stdout:
MAYBE
Problem:
qsort(nil()) -> nil()
qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
lowers(x,nil()) -> nil()
lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
greaters(x,nil()) -> nil()
greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.59 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.59 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ qsort(nil()) -> nil()
, qsort(.(x, y)) ->
++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
, lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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: qsort^#(nil()) -> c_0()
, 2: qsort^#(.(x, y)) ->
c_1(qsort^#(lowers(x, y)), qsort^#(greaters(x, y)))
, 3: lowers^#(x, nil()) -> c_2()
, 4: lowers^#(x, .(y, z)) -> c_3(lowers^#(x, z), lowers^#(x, z))
, 5: greaters^#(x, nil()) -> c_4()
, 6: greaters^#(x, .(y, z)) ->
c_5(greaters^#(x, z), greaters^#(x, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ MAYBE ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: NA
-----------------
The usable rules for this path are:
{ lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {1, 2}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
.(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
[0 1 3] [0 0 0] [0]
[0 0 1] [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]
lowers(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]
greaters(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]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
<=(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]
qsort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(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]
lowers^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
greaters^#(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_4() = [0]
[0]
[0]
c_5(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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{lowers^#(x, .(y, z)) -> c_3(lowers^#(x, z), lowers^#(x, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {4}->{3}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {1, 2}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
++(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]
lowers(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]
greaters(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]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
<=(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]
qsort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(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]
lowers^#(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, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
greaters^#(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_4() = [0]
[0]
[0]
c_5(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]
We have not generated a proof for the resulting sub-problem.
* Path {6}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
.(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
[0 1 3] [0 0 0] [0]
[0 0 1] [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]
lowers(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]
greaters(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]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
<=(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]
qsort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(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]
lowers^#(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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
greaters^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_4() = [0]
[0]
[0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
++(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]
lowers(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]
greaters(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]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
<=(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]
qsort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(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]
lowers^#(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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
greaters^#(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_4() = [0]
[0]
[0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
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: qsort^#(nil()) -> c_0()
, 2: qsort^#(.(x, y)) ->
c_1(qsort^#(lowers(x, y)), qsort^#(greaters(x, y)))
, 3: lowers^#(x, nil()) -> c_2()
, 4: lowers^#(x, .(y, z)) -> c_3(lowers^#(x, z), lowers^#(x, z))
, 5: greaters^#(x, nil()) -> c_4()
, 6: greaters^#(x, .(y, z)) ->
c_5(greaters^#(x, z), greaters^#(x, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ MAYBE ]
|
`->{5} [ NA ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: NA
-----------------
The usable rules for this path are:
{ lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {1, 2}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
.(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
++(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lowers(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
<=(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
qsort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lowers^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
greaters^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {1, 2}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
.(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]
lowers(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
<=(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
qsort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lowers^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
greaters^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
.(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
++(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lowers(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
<=(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
qsort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lowers^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{greaters^#(x, .(y, z)) -> c_5(greaters^#(x, z), greaters^#(x, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
.(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]
lowers(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
<=(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
qsort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lowers^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
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: qsort^#(nil()) -> c_0()
, 2: qsort^#(.(x, y)) ->
c_1(qsort^#(lowers(x, y)), qsort^#(greaters(x, y)))
, 3: lowers^#(x, nil()) -> c_2()
, 4: lowers^#(x, .(y, z)) -> c_3(lowers^#(x, z), lowers^#(x, z))
, 5: greaters^#(x, nil()) -> c_4()
, 6: greaters^#(x, .(y, z)) ->
c_5(greaters^#(x, z), greaters^#(x, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ MAYBE ]
|
`->{5} [ NA ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: NA
-----------------
The usable rules for this path are:
{ lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {1, 2}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [1] x1 + [0] x2 + [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
lowers(x1, x2) = [0] x1 + [0] x2 + [0]
greaters(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
<=(x1, x2) = [0] x1 + [0] x2 + [0]
qsort^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
lowers^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1, x2) = [1] x1 + [1] x2 + [0]
greaters^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {1, 2}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [0] x1 + [0] x2 + [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
lowers(x1, x2) = [0] x1 + [0] x2 + [0]
greaters(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
<=(x1, x2) = [0] x1 + [0] x2 + [0]
qsort^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
lowers^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1, x2) = [1] x1 + [1] x2 + [0]
greaters^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [1] x1 + [0] x2 + [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
lowers(x1, x2) = [0] x1 + [0] x2 + [0]
greaters(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
<=(x1, x2) = [0] x1 + [0] x2 + [0]
qsort^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
lowers^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
greaters^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{greaters^#(x, .(y, z)) -> c_5(greaters^#(x, z), greaters^#(x, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [0] x1 + [0] x2 + [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
lowers(x1, x2) = [0] x1 + [0] x2 + [0]
greaters(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
<=(x1, x2) = [0] x1 + [0] x2 + [0]
qsort^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
lowers^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
greaters^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
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 4.59 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.59 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ qsort(nil()) -> nil()
, qsort(.(x, y)) ->
++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
, lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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: qsort^#(nil()) -> c_0()
, 2: qsort^#(.(x, y)) ->
c_1(qsort^#(lowers(x, y)), x, qsort^#(greaters(x, y)))
, 3: lowers^#(x, nil()) -> c_2()
, 4: lowers^#(x, .(y, z)) ->
c_3(y, x, y, lowers^#(x, z), lowers^#(x, z))
, 5: greaters^#(x, nil()) -> c_4()
, 6: greaters^#(x, .(y, z)) ->
c_5(y, x, greaters^#(x, z), y, greaters^#(x, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ NA ]
|
`->{5} [ NA ]
->{4} [ MAYBE ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: NA
-----------------
The usable rules for this path are:
{ lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {4, 5}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
.(x1, x2) = [1 3 3] x1 + [1 3 3] x2 + [0]
[0 1 3] [0 1 3] [0]
[0 0 1] [0 0 1] [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]
lowers(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]
greaters(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]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
<=(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]
qsort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
lowers^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [1 0 0] x4 + [1 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 1 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0]
greaters^#(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_4() = [0]
[0]
[0]
c_5(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 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: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{lowers^#(x, .(y, z)) ->
c_3(y, x, y, lowers^#(x, z), lowers^#(x, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {4}->{3}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {4, 5}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
++(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]
lowers(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]
greaters(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]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
<=(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]
qsort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
lowers^#(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, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [1 0 0] x4 + [1 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 1 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0]
greaters^#(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_4() = [0]
[0]
[0]
c_5(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {3, 5}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
.(x1, x2) = [1 3 3] x1 + [1 3 3] x2 + [0]
[0 1 3] [0 1 3] [0]
[0 0 1] [0 0 1] [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]
lowers(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]
greaters(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]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
<=(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]
qsort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
lowers^#(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, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
greaters^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_4() = [0]
[0]
[0]
c_5(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0 0 0] x4 + [1 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 1] [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {3, 5}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
++(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]
lowers(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]
greaters(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]
if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
<=(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]
qsort^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
lowers^#(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, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
greaters^#(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_4() = [0]
[0]
[0]
c_5(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0 0 0] x4 + [1 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 1] [0]
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: qsort^#(nil()) -> c_0()
, 2: qsort^#(.(x, y)) ->
c_1(qsort^#(lowers(x, y)), x, qsort^#(greaters(x, y)))
, 3: lowers^#(x, nil()) -> c_2()
, 4: lowers^#(x, .(y, z)) ->
c_3(y, x, y, lowers^#(x, z), lowers^#(x, z))
, 5: greaters^#(x, nil()) -> c_4()
, 6: greaters^#(x, .(y, z)) ->
c_5(y, x, greaters^#(x, z), y, greaters^#(x, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ MAYBE ]
|
`->{5} [ NA ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: NA
-----------------
The usable rules for this path are:
{ lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {4, 5}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
.(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
++(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lowers(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
<=(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
qsort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
lowers^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1 0] x4 + [1 0] x5 + [0]
[0 0] [0 0] [0 0] [0 1] [0 1] [0]
greaters^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {4, 5}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
.(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]
lowers(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
<=(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
qsort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
lowers^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1 0] x4 + [1 0] x5 + [0]
[0 0] [0 0] [0 0] [0 1] [0 1] [0]
greaters^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {3, 5}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
.(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
++(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lowers(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
<=(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
qsort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
lowers^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
greaters^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [1 0] x5 + [0]
[0 0] [0 0] [0 1] [0 0] [0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{greaters^#(x, .(y, z)) ->
c_5(y, x, greaters^#(x, z), y, greaters^#(x, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {3, 5}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
.(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]
lowers(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
greaters(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
<=(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
qsort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
lowers^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
greaters^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [1 0] x5 + [0]
[0 0] [0 0] [0 1] [0 0] [0 1] [0]
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: qsort^#(nil()) -> c_0()
, 2: qsort^#(.(x, y)) ->
c_1(qsort^#(lowers(x, y)), x, qsort^#(greaters(x, y)))
, 3: lowers^#(x, nil()) -> c_2()
, 4: lowers^#(x, .(y, z)) ->
c_3(y, x, y, lowers^#(x, z), lowers^#(x, z))
, 5: greaters^#(x, nil()) -> c_4()
, 6: greaters^#(x, .(y, z)) ->
c_5(y, x, greaters^#(x, z), y, greaters^#(x, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ MAYBE ]
|
`->{5} [ NA ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ inherited ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{1}.
* Path {2}->{1}: NA
-----------------
The usable rules for this path are:
{ lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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}: NA
------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {4, 5}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
lowers(x1, x2) = [0] x1 + [0] x2 + [0]
greaters(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
<=(x1, x2) = [0] x1 + [0] x2 + [0]
qsort^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
lowers^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [1] x4 + [1] x5 + [0]
greaters^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {4, 5}, Uargs(greaters^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [0] x1 + [0] x2 + [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
lowers(x1, x2) = [0] x1 + [0] x2 + [0]
greaters(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
<=(x1, x2) = [0] x1 + [0] x2 + [0]
qsort^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
lowers^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [1] x4 + [1] x5 + [0]
greaters^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {6}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {3, 5}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
lowers(x1, x2) = [0] x1 + [0] x2 + [0]
greaters(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
<=(x1, x2) = [0] x1 + [0] x2 + [0]
qsort^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
lowers^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
greaters^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [1] x3 + [0] x4 + [1] x5 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{greaters^#(x, .(y, z)) ->
c_5(y, x, greaters^#(x, z), y, greaters^#(x, z))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {6}->{5}: NA
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(qsort) = {}, Uargs(.) = {}, Uargs(++) = {},
Uargs(lowers) = {}, Uargs(greaters) = {}, Uargs(if) = {},
Uargs(<=) = {}, Uargs(qsort^#) = {}, Uargs(c_1) = {},
Uargs(lowers^#) = {}, Uargs(c_3) = {}, Uargs(greaters^#) = {},
Uargs(c_5) = {3, 5}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
qsort(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [0] x1 + [0] x2 + [0]
++(x1, x2) = [0] x1 + [0] x2 + [0]
lowers(x1, x2) = [0] x1 + [0] x2 + [0]
greaters(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
<=(x1, x2) = [0] x1 + [0] x2 + [0]
qsort^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
lowers^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
greaters^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [1] x3 + [0] x4 + [1] x5 + [0]
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 4.59 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ qsort(nil()) -> nil()
, qsort(.(x, y)) ->
++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
, lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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 4.59 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ qsort(nil()) -> nil()
, qsort(.(x, y)) ->
++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
, lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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 pair3rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.59 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ qsort(nil()) -> nil()
, qsort(.(x, y)) ->
++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
, lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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 4.59 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ qsort(nil()) -> nil()
, qsort(.(x, y)) ->
++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
, lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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 | 69.59177ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.59 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ qsort(nil()) -> nil()
, qsort(.(x, y)) ->
++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
, lowers(x, nil()) -> nil()
, lowers(x, .(y, z)) ->
if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
, greaters(x, nil()) -> nil()
, greaters(x, .(y, z)) ->
if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))}
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..