Tool CaT
stdout:
MAYBE
Problem:
g(X) -> u(h(X),h(X),X)
u(d(),c(Y),X) -> k(Y)
h(d()) -> c(a())
h(d()) -> c(b())
f(k(a()),k(b()),X) -> f(X,X,X)
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:
{ g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)
, h(d()) -> c(a())
, h(d()) -> c(b())
, f(k(a()), k(b()), X) -> f(X, X, X)}
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: g^#(X) -> c_0(u^#(h(X), h(X), X))
, 2: u^#(d(), c(Y), X) -> c_1()
, 3: h^#(d()) -> c_2()
, 4: h^#(d()) -> c_3()
, 5: f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ MAYBE ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {1},
Uargs(u^#) = {1, 2}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [3]
[0 0 0] [3]
[0 0 0] [3]
d() = [0]
[0]
[0]
c(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
u^#(x1, x2, x3) = [3 2 0] x1 + [2 2 2] 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() = [0]
[0]
[0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [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_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 have not generated a proof for the resulting sub-problem.
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d() = [0]
[0]
[0]
c(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(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() = [0]
[0]
[0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {u^#(d(), c(Y), X) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(c) = {}, Uargs(u^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
[2]
c(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
u^#(x1, x2, x3) = [2 2 2] x1 + [0 0 0] x2 + [0 0 0] x3 + [3]
[0 0 2] [2 0 0] [0 0 0] [7]
[0 0 0] [0 2 2] [0 0 0] [7]
c_1() = [0]
[1]
[1]
* Path {3}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d() = [0]
[0]
[0]
c(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(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() = [0]
[0]
[0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
[2]
h^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_2() = [0]
[1]
[1]
* 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d() = [0]
[0]
[0]
c(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(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() = [0]
[0]
[0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
[2]
h^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d() = [0]
[0]
[0]
c(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(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() = [0]
[0]
[0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_4(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: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(X) -> c_0(u^#(h(X), h(X), X))
, 2: u^#(d(), c(Y), X) -> c_1()
, 3: h^#(d()) -> c_2()
, 4: h^#(d()) -> c_3()
, 5: f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ MAYBE ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {1},
Uargs(u^#) = {1, 2}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [3]
[0 0] [3]
d() = [0]
[0]
c(x1) = [1 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [1]
[0]
b() = [1]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
u^#(x1, x2, x3) = [1 0] x1 + [1 3] x2 + [0 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_1() = [0]
[0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {u^#(d(), c(Y), X) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(c) = {}, Uargs(u^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
c(x1) = [0 0] x1 + [2]
[0 0] [0]
u^#(x1, x2, x3) = [2 2] x1 + [2 0] x2 + [0 0] x3 + [3]
[0 0] [0 0] [0 0] [3]
c_1() = [0]
[1]
* Path {3}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
h^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
* 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
h^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_4(x1) = [1 0] x1 + [0]
[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: {f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(X) -> c_0(u^#(h(X), h(X), X))
, 2: u^#(d(), c(Y), X) -> c_1()
, 3: h^#(d()) -> c_2()
, 4: h^#(d()) -> c_3()
, 5: f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ MAYBE ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {1},
Uargs(u^#) = {1, 2}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [1] x1 + [2]
d() = [1]
c(x1) = [0] x1 + [2]
k(x1) = [0] x1 + [0]
a() = [3]
b() = [3]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
u^#(x1, x2, x3) = [1] x1 + [1] x2 + [0] x3 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(1))
We have not generated a proof for the resulting sub-problem.
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [0] x1 + [0]
d() = [0]
c(x1) = [0] x1 + [0]
k(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
u^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {u^#(d(), c(Y), X) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(c) = {}, Uargs(u^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
c(x1) = [0] x1 + [2]
u^#(x1, x2, x3) = [2] x1 + [2] x2 + [0] x3 + [7]
c_1() = [0]
* Path {3}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [0] x1 + [0]
d() = [0]
c(x1) = [0] x1 + [0]
k(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
u^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [7]
h^#(x1) = [1] x1 + [7]
c_2() = [1]
* 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [0] x1 + [0]
d() = [0]
c(x1) = [0] x1 + [0]
k(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
u^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [7]
h^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(h^#) = {}, Uargs(f^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [0] x1 + [0]
d() = [0]
c(x1) = [0] x1 + [0]
k(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
u^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
c_4(x1) = [1] x1 + [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: {f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
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:
{ g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)
, h(d()) -> c(a())
, h(d()) -> c(b())
, f(k(a()), k(b()), X) -> f(X, X, X)}
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: g^#(X) -> c_0(u^#(h(X), h(X), X))
, 2: u^#(d(), c(Y), X) -> c_1(Y)
, 3: h^#(d()) -> c_2()
, 4: h^#(d()) -> c_3()
, 5: f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ MAYBE ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {1},
Uargs(u^#) = {1, 2}, Uargs(c_1) = {}, Uargs(h^#) = {},
Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [3]
[0 0 0] [3]
[0 0 0] [3]
d() = [0]
[0]
[0]
c(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
u^#(x1, x2, x3) = [3 2 0] x1 + [2 2 2] 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) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [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_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 have not generated a proof for the resulting sub-problem.
* Path {2}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d() = [0]
[0]
[0]
c(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] 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) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [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_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(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {u^#(d(), c(Y), X) -> c_1(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(c) = {}, Uargs(u^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[3]
[0]
c(x1) = [1 2 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
u^#(x1, x2, x3) = [2 2 0] x1 + [2 2 1] x2 + [0 0 0] x3 + [3]
[0 2 0] [2 0 2] [0 0 0] [3]
[2 0 0] [0 0 0] [0 0 0] [7]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {3}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d() = [0]
[0]
[0]
c(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(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]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [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_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: {h^#(d()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
[2]
h^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_2() = [0]
[1]
[1]
* 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d() = [0]
[0]
[0]
c(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(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]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [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_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: {h^#(d()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
[2]
h^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u(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]
h(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d() = [0]
[0]
[0]
c(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
k(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
b() = [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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(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]
h^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_4(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: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(X) -> c_0(u^#(h(X), h(X), X))
, 2: u^#(d(), c(Y), X) -> c_1(Y)
, 3: h^#(d()) -> c_2()
, 4: h^#(d()) -> c_3()
, 5: f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ MAYBE ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {1},
Uargs(u^#) = {1, 2}, Uargs(c_1) = {}, Uargs(h^#) = {},
Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [3]
[0 0] [3]
d() = [0]
[0]
c(x1) = [1 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [1]
[0]
b() = [1]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
u^#(x1, x2, x3) = [1 0] x1 + [1 3] x2 + [0 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [1 1] x1 + [0]
[0 0] [0]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {u^#(d(), c(Y), X) -> c_1(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(c) = {}, Uargs(u^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
c(x1) = [1 2] x1 + [0]
[0 0] [2]
u^#(x1, x2, x3) = [2 2] x1 + [0 2] x2 + [0 0] x3 + [3]
[2 2] [2 2] [0 0] [3]
c_1(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {3}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(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]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
h^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
* 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(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]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
[2]
h^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0 0] x1 + [0]
[0 0] [0]
u(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
d() = [0]
[0]
c(x1) = [0 0] x1 + [0]
[0 0] [0]
k(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
b() = [0]
[0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(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]
h^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_4(x1) = [1 0] x1 + [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: {f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(X) -> c_0(u^#(h(X), h(X), X))
, 2: u^#(d(), c(Y), X) -> c_1(Y)
, 3: h^#(d()) -> c_2()
, 4: h^#(d()) -> c_3()
, 5: f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ MAYBE ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ h(d()) -> c(a())
, h(d()) -> c(b())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {1},
Uargs(u^#) = {1, 2}, Uargs(c_1) = {}, Uargs(h^#) = {},
Uargs(f^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [1] x1 + [2]
d() = [1]
c(x1) = [0] x1 + [2]
k(x1) = [0] x1 + [0]
a() = [3]
b() = [3]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
u^#(x1, x2, x3) = [1] x1 + [1] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(1))
We have not generated a proof for the resulting sub-problem.
* Path {2}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [0] x1 + [0]
d() = [0]
c(x1) = [0] x1 + [0]
k(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
u^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
c_1(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {u^#(d(), c(Y), X) -> c_1(Y)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(c) = {}, Uargs(u^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [2]
c(x1) = [1] x1 + [2]
u^#(x1, x2, x3) = [2] x1 + [2] x2 + [0] x3 + [7]
c_1(x1) = [0] x1 + [0]
* Path {3}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [0] x1 + [0]
d() = [0]
c(x1) = [0] x1 + [0]
k(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
u^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [7]
h^#(x1) = [1] x1 + [7]
c_2() = [1]
* 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [0] x1 + [0]
d() = [0]
c(x1) = [0] x1 + [0]
k(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
u^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {h^#(d()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(h^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
d() = [7]
h^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: 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(g) = {}, Uargs(u) = {}, Uargs(h) = {}, Uargs(c) = {},
Uargs(k) = {}, Uargs(f) = {}, Uargs(g^#) = {}, Uargs(c_0) = {},
Uargs(u^#) = {}, Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(f^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
u(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
h(x1) = [0] x1 + [0]
d() = [0]
c(x1) = [0] x1 + [0]
k(x1) = [0] x1 + [0]
a() = [0]
b() = [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
u^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
c_4(x1) = [1] x1 + [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: {f^#(k(a()), k(b()), X) -> c_4(f^#(X, X, X))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
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.