Tool CaT
stdout:
MAYBE
Problem:
h(X,Z) -> f(X,s(X),Z)
f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
g(0(),Y) -> 0()
g(X,s(Y)) -> g(X,Y)
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ h(X, Z) -> f(X, s(X), Z)
, f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, 2: f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))
, 3: g^#(0(), Y) -> c_2()
, 4: g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^3)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {1}, Uargs(f^#) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
f(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [2]
g(x1, x2) = [2 0 0] x1 + [0 0 1] x2 + [0]
[0 0 0] [0 0 0] [0]
[3 0 0] [0 0 0] [0]
0() = [3]
[0]
[0]
h^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
g^#(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]
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: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))}
Weak Rules:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
f(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]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
g(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]
h^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [3 3 3] 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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 1] [2]
g^#(x1, x2) = [0 0 0] x1 + [0 1 2] x2 + [2]
[4 4 4] [2 2 2] [0]
[4 4 4] [2 0 2] [0]
c_3(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* Path {4}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
f(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(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]
h^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
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: {g^#(0(), Y) -> c_2()}
Weak Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 1 2] [0]
[0 0 0] [3]
0() = [2]
[2]
[2]
g^#(x1, x2) = [2 0 2] x1 + [0 4 3] x2 + [0]
[2 2 2] [4 0 2] [2]
[0 2 2] [4 0 0] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, 2: f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))
, 3: g^#(0(), Y) -> c_2()
, 4: g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {1}, Uargs(f^#) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 1] [2]
g(x1, x2) = [2 0] x1 + [0 1] x2 + [0]
[0 0] [0 0] [0]
0() = [2]
[0]
h^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
g^#(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]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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:
{ h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))}
Weak Rules:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
h^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
g^#(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[4 4] [4 5] [0]
c_3(x1) = [1 0] x1 + [1]
[0 0] [7]
* Path {4}->{3}: YES(?,O(n^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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
h^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(0(), Y) -> c_2()}
Weak Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 0] [2]
0() = [0]
[2]
g^#(x1, x2) = [0 2] x1 + [0 0] x2 + [0]
[4 0] [2 2] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, 2: f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))
, 3: g^#(0(), Y) -> c_2()
, 4: g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {1}, Uargs(f^#) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [3]
g(x1, x2) = [0] x1 + [2] x2 + [3]
0() = [2]
h^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [1] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [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 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))}
Weak Rules:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
h^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
g^#(x1, x2) = [7] x1 + [2] x2 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{3}: YES(?,O(n^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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
h^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(0(), Y) -> c_2()}
Weak Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
g^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_2() = [1]
c_3(x1) = [1] x1 + [4]
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
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ h(X, Z) -> f(X, s(X), Z)
, f(X, Y, g(X, Y)) -> h(0(), g(X, Y))
, g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, 2: f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))
, 3: g^#(0(), Y) -> c_2()
, 4: g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^3)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {1}, Uargs(f^#) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
f(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [2]
g(x1, x2) = [2 0 0] x1 + [0 0 1] x2 + [0]
[0 0 0] [0 0 0] [0]
[3 0 0] [0 0 0] [0]
0() = [3]
[0]
[0]
h^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
g^#(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]
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: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))}
Weak Rules:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
f(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]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
g(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]
h^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1, x2) = [3 3 3] 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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 1] [2]
g^#(x1, x2) = [0 0 0] x1 + [0 1 2] x2 + [2]
[4 4 4] [2 2 2] [0]
[4 4 4] [2 0 2] [0]
c_3(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* Path {4}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(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]
f(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(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]
h^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
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: {g^#(0(), Y) -> c_2()}
Weak Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 1 2] [0]
[0 0 0] [3]
0() = [2]
[2]
[2]
g^#(x1, x2) = [2 0 2] x1 + [0 4 3] x2 + [0]
[2 2 2] [4 0 2] [2]
[0 2 2] [4 0 0] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, 2: f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))
, 3: g^#(0(), Y) -> c_2()
, 4: g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {1}, Uargs(f^#) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 1] [2]
g(x1, x2) = [2 0] x1 + [0 1] x2 + [0]
[0 0] [0 0] [0]
0() = [2]
[0]
h^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
g^#(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]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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:
{ h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))}
Weak Rules:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
h^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
g^#(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[4 4] [4 5] [0]
c_3(x1) = [1 0] x1 + [1]
[0 0] [7]
* Path {4}->{3}: YES(?,O(n^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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
h^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(0(), Y) -> c_2()}
Weak Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 0] [2]
0() = [0]
[2]
g^#(x1, x2) = [0 2] x1 + [0 0] x2 + [0]
[4 0] [2 2] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, 2: f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))
, 3: g^#(0(), Y) -> c_2()
, 4: g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {1}, Uargs(f^#) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [3]
g(x1, x2) = [0] x1 + [2] x2 + [3]
0() = [2]
h^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [1] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [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 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ h^#(X, Z) -> c_0(f^#(X, s(X), Z))
, f^#(X, Y, g(X, Y)) -> c_1(h^#(0(), g(X, Y)))}
Weak Rules:
{ g(0(), Y) -> 0()
, g(X, s(Y)) -> g(X, Y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
h^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
g^#(x1, x2) = [7] x1 + [2] x2 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{3}: YES(?,O(n^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(h) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {},
Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
h(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
h^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(0(), Y) -> c_2()}
Weak Rules: {g^#(X, s(Y)) -> c_3(g^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
g^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_2() = [1]
c_3(x1) = [1] x1 + [4]
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.