Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.13 |
|---|
stdout:
MAYBE
Problem:
-(0(),y) -> 0()
-(x,0()) -> x
-(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
p(0()) -> 0()
p(s(x)) -> x
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.13 |
|---|
stdout:
YES(?,O(n^1))
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^3)) |
|---|
| Input | SK90 4.13 |
|---|
stdout:
YES(?,O(n^3))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^3))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: -^#(0(), y) -> c_0()
, 2: -^#(x, 0()) -> c_1()
, 3: -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, 4: p^#(0()) -> c_3()
, 5: p^#(s(x)) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^3)) ]
|
`->{2} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {2}, Uargs(c_2) = {1}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 1 1] x1 + [0]
[0 1 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]
greater(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]
p(x1) = [1 0 0] x1 + [2]
[0 2 0] [0]
[1 0 0] [0]
-^#(x1, x2) = [3 3 3] x1 + [1 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {-^#(x, s(y)) -> c_2(-^#(x, p(s(y))))}
Weak Rules:
{ p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 2 1] x1 + [0]
[0 0 1] [0]
[0 0 1] [4]
p(x1) = [4 0 0] x1 + [0]
[2 0 0] [4]
[0 1 0] [0]
-^#(x1, x2) = [0 0 0] x1 + [0 0 2] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 4 4] [3 1 0] [0]
c_2(x1) = [1 0 0] x1 + [5]
[0 0 0] [0]
[0 0 0] [0]
* Path {3}->{1}: YES(?,O(n^3))
----------------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {2}, Uargs(c_2) = {1}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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]
greater(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]
p(x1) = [2 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
-^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {-^#(0(), y) -> c_0()}
Weak Rules:
{ -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [2]
p(x1) = [1 0 0] x1 + [0]
[0 1 0] [4]
[0 0 4] [0]
-^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 2] [4]
[4 4 0] [0 0 0] [4]
c_0() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [7]
[2 0 0] [0]
* Path {3}->{2}: YES(?,O(n^3))
----------------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {2}, Uargs(c_2) = {1}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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]
greater(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]
p(x1) = [2 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
-^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {-^#(x, 0()) -> c_1()}
Weak Rules:
{ -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 2 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [0]
p(x1) = [1 0 0] x1 + [0]
[4 0 0] [0]
[0 0 2] [0]
-^#(x1, x2) = [0 0 0] x1 + [2 0 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[4 4 4] [0 7 1] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 2 0] [0]
* Path {4}: 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(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_2) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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]
greater(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]
p(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_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [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: {p^#(0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
p^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: 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(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_2) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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]
greater(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]
p(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_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [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: {p^#(s(x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
p^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_4() = [0]
[1]
[1]Tool RC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | SK90 4.13 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^3)) |
|---|
| Input | SK90 4.13 |
|---|
stdout:
YES(?,O(n^3))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^3))
Input Problem: runtime-complexity with respect to
Rules:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^3))
Input Problem: runtime-complexity with respect to
Rules:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: -^#(0(), y) -> c_0()
, 2: -^#(x, 0()) -> c_1(x)
, 3: -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y))))
, 4: p^#(0()) -> c_3()
, 5: p^#(s(x)) -> c_4(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^3)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^3)) ]
|
`->{2} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {2}, Uargs(c_1) = {}, Uargs(c_2) = {3},
Uargs(p^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 2 1] x1 + [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]
greater(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]
p(x1) = [1 0 0] x1 + [1]
[2 0 0] [0]
[2 0 0] [0]
-^#(x1, x2) = [3 3 2] x1 + [1 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {-^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y))))}
Weak Rules:
{ p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_2) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
p(x1) = [4 0 0] x1 + [4]
[1 0 0] [4]
[0 1 0] [0]
-^#(x1, x2) = [0 0 0] x1 + [0 0 4] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [5]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
* Path {3}->{1}: YES(?,O(n^3))
----------------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {2}, Uargs(c_1) = {}, Uargs(c_2) = {3},
Uargs(p^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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]
greater(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]
p(x1) = [2 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
-^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {-^#(0(), y) -> c_0()}
Weak Rules:
{ -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y))))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_2) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [4]
[2]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [2]
p(x1) = [1 0 2] x1 + [0]
[1 0 2] [0]
[0 0 2] [0]
-^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 4 0] [0 0 4] [0]
c_0() = [1]
[0]
[0]
c_2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 4] [2 0 0] [3]
* Path {3}->{2}: YES(?,O(n^3))
----------------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {2}, Uargs(c_1) = {}, Uargs(c_2) = {3},
Uargs(p^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[3]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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]
greater(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]
p(x1) = [2 0 0] x1 + [2]
[0 2 0] [2]
[0 0 1] [0]
-^#(x1, x2) = [3 3 3] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {-^#(x, 0()) -> c_1(x)}
Weak Rules:
{ -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y))))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 4] [0]
[0 0 0] [0]
p(x1) = [4 0 0] x1 + [4]
[0 1 0] [4]
[0 1 0] [0]
-^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [1]
[0 0 2] [0 0 0] [0]
[0 0 2] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 1] [0 0 0] [0 0 0] [0]
[0 0 1] [0 0 0] [0 0 0] [0]
* Path {4}: 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(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(p^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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]
greater(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]
p(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_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
p^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(-) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(greater) = {},
Uargs(p) = {}, Uargs(-^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(p^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
-(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 3] x1 + [0]
[0 1 1] [0]
[0 0 1] [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]
greater(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]
p(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_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(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]
p^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
p^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.13 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
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 pair3irc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.13 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
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 4.13 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
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 4.13 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
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 | 61.510876ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | SK90 4.13 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ -(0(), y) -> 0()
, -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, p(0()) -> 0()
, p(s(x)) -> x}
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..