Tool CaT
stdout:
MAYBE
Problem:
f(g(x),h(x)) -> f(i(x),h(x))
f(x,h(y)) -> f(x,j(y))
f(i(x),h(x)) -> a()
f(x,j(y)) -> a()
i(x) -> g(x)
j(x) -> h(x)
Proof:
Complexity Transformation Processor:
strict:
f(g(x),h(x)) -> f(i(x),h(x))
f(x,h(y)) -> f(x,j(y))
f(i(x),h(x)) -> a()
f(x,j(y)) -> a()
i(x) -> g(x)
j(x) -> h(x)
weak:
Matrix Interpretation Processor:
dimension: 3
max_matrix:
[1 2 1]
[0 0 2]
[0 0 0]
interpretation:
[0]
[a] = [0]
[0],
[1 1 0]
[j](x0) = [0 0 0]x0
[0 0 0] ,
[1 2 0] [0]
[i](x0) = [0 0 2]x0 + [0]
[0 0 0] [2],
[1 0 1] [1 0 0] [0]
[f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [3]
[0 0 0] [0 0 0] [0],
[1 1 0]
[h](x0) = [0 0 0]x0
[0 0 0] ,
[1 2 0] [0]
[g](x0) = [0 0 0]x0 + [0]
[0 0 0] [2]
orientation:
[2 3 0] [2] [2 3 0] [2]
f(g(x),h(x)) = [0 0 0]x + [3] >= [0 0 0]x + [3] = f(i(x),h(x))
[0 0 0] [0] [0 0 0] [0]
[1 0 1] [1 1 0] [0] [1 0 1] [1 1 0] [0]
f(x,h(y)) = [0 0 0]x + [0 0 0]y + [3] >= [0 0 0]x + [0 0 0]y + [3] = f(x,j(y))
[0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0]
[2 3 0] [2] [0]
f(i(x),h(x)) = [0 0 0]x + [3] >= [0] = a()
[0 0 0] [0] [0]
[1 0 1] [1 1 0] [0] [0]
f(x,j(y)) = [0 0 0]x + [0 0 0]y + [3] >= [0] = a()
[0 0 0] [0 0 0] [0] [0]
[1 2 0] [0] [1 2 0] [0]
i(x) = [0 0 2]x + [0] >= [0 0 0]x + [0] = g(x)
[0 0 0] [2] [0 0 0] [2]
[1 1 0] [1 1 0]
j(x) = [0 0 0]x >= [0 0 0]x = h(x)
[0 0 0] [0 0 0]
problem:
strict:
f(g(x),h(x)) -> f(i(x),h(x))
f(x,h(y)) -> f(x,j(y))
f(x,j(y)) -> a()
i(x) -> g(x)
j(x) -> h(x)
weak:
f(i(x),h(x)) -> a()
Matrix Interpretation Processor:
dimension: 4
max_matrix:
[1 0 2 1]
[0 0 3 2]
[0 0 0 1]
[0 0 0 0]
interpretation:
[0]
[3]
[a] = [0]
[0],
[1 0 0 1] [0]
[0 0 2 1] [0]
[j](x0) = [0 0 0 1]x0 + [1]
[0 0 0 0] [1],
[1 0 2 0] [0]
[0 0 2 2] [1]
[i](x0) = [0 0 0 0]x0 + [0]
[0 0 0 0] [0],
[1 0 0 0] [1 0 2 1]
[0 0 0 0] [0 0 3 0]
[f](x0, x1) = [0 0 0 0]x0 + [0 0 0 0]x1
[0 0 0 0] [0 0 0 0] ,
[1 0 0 1] [0]
[0 0 0 0] [0]
[h](x0) = [0 0 0 1]x0 + [1]
[0 0 0 0] [1],
[1 0 2 0]
[0 0 0 0]
[g](x0) = [0 0 0 0]x0
[0 0 0 0]
orientation:
[2 0 2 3] [3] [2 0 2 3] [3]
[0 0 0 3] [3] [0 0 0 3] [3]
f(g(x),h(x)) = [0 0 0 0]x + [0] >= [0 0 0 0]x + [0] = f(i(x),h(x))
[0 0 0 0] [0] [0 0 0 0] [0]
[1 0 0 0] [1 0 0 3] [3] [1 0 0 0] [1 0 0 3] [3]
[0 0 0 0] [0 0 0 3] [3] [0 0 0 0] [0 0 0 3] [3]
f(x,h(y)) = [0 0 0 0]x + [0 0 0 0]y + [0] >= [0 0 0 0]x + [0 0 0 0]y + [0] = f(x,j(y))
[0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0]
[1 0 0 0] [1 0 0 3] [3] [0]
[0 0 0 0] [0 0 0 3] [3] [3]
f(x,j(y)) = [0 0 0 0]x + [0 0 0 0]y + [0] >= [0] = a()
[0 0 0 0] [0 0 0 0] [0] [0]
[1 0 2 0] [0] [1 0 2 0]
[0 0 2 2] [1] [0 0 0 0]
i(x) = [0 0 0 0]x + [0] >= [0 0 0 0]x = g(x)
[0 0 0 0] [0] [0 0 0 0]
[1 0 0 1] [0] [1 0 0 1] [0]
[0 0 2 1] [0] [0 0 0 0] [0]
j(x) = [0 0 0 1]x + [1] >= [0 0 0 1]x + [1] = h(x)
[0 0 0 0] [1] [0 0 0 0] [1]
[2 0 2 3] [3] [0]
[0 0 0 3] [3] [3]
f(i(x),h(x)) = [0 0 0 0]x + [0] >= [0] = a()
[0 0 0 0] [0] [0]
problem:
strict:
f(g(x),h(x)) -> f(i(x),h(x))
f(x,h(y)) -> f(x,j(y))
i(x) -> g(x)
j(x) -> h(x)
weak:
f(x,j(y)) -> a()
f(i(x),h(x)) -> a()
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[a] = 80,
[j](x0) = x0,
[i](x0) = x0,
[f](x0, x1) = x0 + x1 + 80,
[h](x0) = x0,
[g](x0) = x0 + 16
orientation:
f(g(x),h(x)) = 2x + 96 >= 2x + 80 = f(i(x),h(x))
f(x,h(y)) = x + y + 80 >= x + y + 80 = f(x,j(y))
i(x) = x >= x + 16 = g(x)
j(x) = x >= x = h(x)
f(x,j(y)) = x + y + 80 >= 80 = a()
f(i(x),h(x)) = 2x + 80 >= 80 = a()
problem:
strict:
f(x,h(y)) -> f(x,j(y))
i(x) -> g(x)
j(x) -> h(x)
weak:
f(g(x),h(x)) -> f(i(x),h(x))
f(x,j(y)) -> a()
f(i(x),h(x)) -> a()
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[a] = 131,
[j](x0) = x0 + 128,
[i](x0) = x0 + 79,
[f](x0, x1) = x0 + x1 + 3,
[h](x0) = x0 + 49,
[g](x0) = x0 + 79
orientation:
f(x,h(y)) = x + y + 52 >= x + y + 131 = f(x,j(y))
i(x) = x + 79 >= x + 79 = g(x)
j(x) = x + 128 >= x + 49 = h(x)
f(g(x),h(x)) = 2x + 131 >= 2x + 131 = f(i(x),h(x))
f(x,j(y)) = x + y + 131 >= 131 = a()
f(i(x),h(x)) = 2x + 131 >= 131 = a()
problem:
strict:
f(x,h(y)) -> f(x,j(y))
i(x) -> g(x)
weak:
j(x) -> h(x)
f(g(x),h(x)) -> f(i(x),h(x))
f(x,j(y)) -> a()
f(i(x),h(x)) -> a()
Open
Tool IRC1
stdout:
MAYBE
Warning when parsing problem:
Unsupported strategy 'OUTERMOST'Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(g(x), h(x)) -> f(i(x), h(x))
, f(x, h(y)) -> f(x, j(y))
, f(i(x), h(x)) -> a()
, f(x, j(y)) -> a()
, i(x) -> g(x)
, j(x) -> h(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: f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, 2: f^#(x, h(y)) -> c_1(f^#(x, j(y)))
, 3: f^#(i(x), h(x)) -> c_2()
, 4: f^#(x, j(y)) -> c_3()
, 5: i^#(x) -> c_4()
, 6: j^#(x) -> c_5()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{1,2} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [3]
h(x1) = [0 1 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [1]
i(x1) = [1 0 0] x1 + [3]
[3 0 0] [3]
[3 0 0] [3]
j(x1) = [0 1 0] x1 + [3]
[3 0 0] [3]
[3 0 0] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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:
{ f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, f^#(x, h(y)) -> c_1(f^#(x, j(y)))}
Weak Rules:
{ i(x) -> g(x)
, j(x) -> h(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,2}->{3}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
h(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
i(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
j(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 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]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {1,2}->{4}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
h(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
i(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
j(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 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]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(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]
i(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
j(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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: {i^#(x) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [0 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
c_4() = [0]
[3]
[3]
* Path {6}: 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(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]
i(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
j(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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: {j^#(x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
j^#(x1) = [0 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
c_5() = [0]
[3]
[3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, 2: f^#(x, h(y)) -> c_1(f^#(x, j(y)))
, 3: f^#(i(x), h(x)) -> c_2()
, 4: f^#(x, j(y)) -> c_3()
, 5: i^#(x) -> c_4()
, 6: j^#(x) -> c_5()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{1,2} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 0] x1 + [1]
[0 0] [1]
h(x1) = [0 1] x1 + [1]
[0 0] [1]
i(x1) = [1 0] x1 + [3]
[3 0] [3]
j(x1) = [0 1] x1 + [3]
[3 0] [3]
a() = [0]
[0]
f^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [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:
{ f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, f^#(x, h(y)) -> c_1(f^#(x, j(y)))}
Weak Rules:
{ i(x) -> g(x)
, j(x) -> h(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,2}->{3}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 1] x1 + [0]
[0 1] [1]
h(x1) = [1 1] x1 + [0]
[0 1] [1]
i(x1) = [3 3] x1 + [3]
[0 3] [3]
j(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {1,2}->{4}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 1] x1 + [0]
[0 1] [1]
h(x1) = [1 1] x1 + [0]
[0 1] [1]
i(x1) = [3 3] x1 + [3]
[0 3] [3]
j(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
i(x1) = [0 0] x1 + [0]
[0 0] [0]
j(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [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: {i^#(x) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [0 0] x1 + [7]
[0 0] [7]
c_4() = [0]
[1]
* Path {6}: 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
i(x1) = [0 0] x1 + [0]
[0 0] [0]
j(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [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: {j^#(x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
j^#(x1) = [0 0] x1 + [7]
[0 0] [7]
c_5() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, 2: f^#(x, h(y)) -> c_1(f^#(x, j(y)))
, 3: f^#(i(x), h(x)) -> c_2()
, 4: f^#(x, j(y)) -> c_3()
, 5: i^#(x) -> c_4()
, 6: j^#(x) -> c_5()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{1,2} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [1]
h(x1) = [1] x1 + [1]
i(x1) = [0] x1 + [3]
j(x1) = [1] x1 + [3]
a() = [0]
f^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4() = [0]
j^#(x1) = [0] x1 + [0]
c_5() = [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:
{ f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, f^#(x, h(y)) -> c_1(f^#(x, j(y)))}
Weak Rules:
{ i(x) -> g(x)
, j(x) -> h(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,2}->{3}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
i(x1) = [3] x1 + [3]
j(x1) = [3] x1 + [3]
a() = [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4() = [0]
j^#(x1) = [0] x1 + [0]
c_5() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {1,2}->{4}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
i(x1) = [3] x1 + [3]
j(x1) = [3] x1 + [3]
a() = [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4() = [0]
j^#(x1) = [0] x1 + [0]
c_5() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
i(x1) = [0] x1 + [0]
j(x1) = [0] x1 + [0]
a() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4() = [0]
j^#(x1) = [0] x1 + [0]
c_5() = [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: {i^#(x) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [0] x1 + [7]
c_4() = [0]
* Path {6}: 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
i(x1) = [0] x1 + [0]
j(x1) = [0] x1 + [0]
a() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4() = [0]
j^#(x1) = [0] x1 + [0]
c_5() = [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: {j^#(x) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(j^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
j^#(x1) = [0] x1 + [7]
c_5() = [0]
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
Warning when parsing problem:
Unsupported strategy 'OUTERMOST'Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f(g(x), h(x)) -> f(i(x), h(x))
, f(x, h(y)) -> f(x, j(y))
, f(i(x), h(x)) -> a()
, f(x, j(y)) -> a()
, i(x) -> g(x)
, j(x) -> h(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: f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, 2: f^#(x, h(y)) -> c_1(f^#(x, j(y)))
, 3: f^#(i(x), h(x)) -> c_2()
, 4: f^#(x, j(y)) -> c_3()
, 5: i^#(x) -> c_4(x)
, 6: j^#(x) -> c_5(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{1,2} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [3]
h(x1) = [0 1 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [1]
i(x1) = [1 0 0] x1 + [3]
[3 0 0] [3]
[3 0 0] [3]
j(x1) = [0 1 0] x1 + [3]
[3 0 0] [3]
[3 0 0] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [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]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(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:
{ f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, f^#(x, h(y)) -> c_1(f^#(x, j(y)))}
Weak Rules:
{ i(x) -> g(x)
, j(x) -> h(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,2}->{3}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
h(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
i(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
j(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 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]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [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]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(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 have not generated a proof for the resulting sub-problem.
* Path {1,2}->{4}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
h(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
i(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
j(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 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]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [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]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(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 have not generated a proof for the resulting sub-problem.
* 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(c_4) = {}, Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(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]
i(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
j(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
j^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(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: {i^#(x) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [7 7 7] x1 + [7]
[7 7 7] [7]
[7 7 7] [7]
c_4(x1) = [3 3 3] x1 + [0]
[3 1 3] [1]
[1 1 1] [1]
* Path {6}: 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(c_4) = {}, Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
g(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]
i(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
j(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
i^#(x1) = [0 0 0] x1 + [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]
j^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [1 1 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(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {j^#(x) -> c_5(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
j^#(x1) = [7 7 7] x1 + [7]
[7 7 7] [7]
[7 7 7] [7]
c_5(x1) = [3 3 3] x1 + [0]
[3 1 3] [1]
[1 1 1] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, 2: f^#(x, h(y)) -> c_1(f^#(x, j(y)))
, 3: f^#(i(x), h(x)) -> c_2()
, 4: f^#(x, j(y)) -> c_3()
, 5: i^#(x) -> c_4(x)
, 6: j^#(x) -> c_5(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{1,2} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 0] x1 + [1]
[0 0] [1]
h(x1) = [0 1] x1 + [1]
[0 0] [1]
i(x1) = [1 0] x1 + [3]
[3 0] [3]
j(x1) = [0 1] x1 + [3]
[3 0] [3]
a() = [0]
[0]
f^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(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:
{ f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, f^#(x, h(y)) -> c_1(f^#(x, j(y)))}
Weak Rules:
{ i(x) -> g(x)
, j(x) -> h(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,2}->{3}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 1] x1 + [0]
[0 1] [1]
h(x1) = [1 1] x1 + [0]
[0 1] [1]
i(x1) = [3 3] x1 + [3]
[0 3] [3]
j(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {1,2}->{4}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [1 1] x1 + [0]
[0 1] [1]
h(x1) = [1 1] x1 + [0]
[0 1] [1]
i(x1) = [3 3] x1 + [3]
[0 3] [3]
j(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(c_4) = {}, Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
i(x1) = [0 0] x1 + [0]
[0 0] [0]
j(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_4(x1) = [1 1] x1 + [0]
[0 0] [0]
j^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(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: {i^#(x) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [7 7] x1 + [7]
[7 7] [7]
c_4(x1) = [1 3] x1 + [0]
[3 1] [3]
* Path {6}: 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(c_4) = {}, Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
h(x1) = [0 0] x1 + [0]
[0 0] [0]
i(x1) = [0 0] x1 + [0]
[0 0] [0]
j(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
i^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
j^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_5(x1) = [1 1] 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: {j^#(x) -> c_5(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
j^#(x1) = [7 7] x1 + [7]
[7 7] [7]
c_5(x1) = [1 3] x1 + [0]
[3 1] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, 2: f^#(x, h(y)) -> c_1(f^#(x, j(y)))
, 3: f^#(i(x), h(x)) -> c_2()
, 4: f^#(x, j(y)) -> c_3()
, 5: i^#(x) -> c_4(x)
, 6: j^#(x) -> c_5(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{1,2} [ MAYBE ]
|
|->{3} [ NA ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [1]
h(x1) = [1] x1 + [1]
i(x1) = [0] x1 + [3]
j(x1) = [1] x1 + [3]
a() = [0]
f^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
j^#(x1) = [0] x1 + [0]
c_5(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:
{ f^#(g(x), h(x)) -> c_0(f^#(i(x), h(x)))
, f^#(x, h(y)) -> c_1(f^#(x, j(y)))}
Weak Rules:
{ i(x) -> g(x)
, j(x) -> h(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,2}->{3}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
i(x1) = [3] x1 + [3]
j(x1) = [3] x1 + [3]
a() = [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
j^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {1,2}->{4}: NA
-------------------
The usable rules for this path are:
{ i(x) -> g(x)
, j(x) -> h(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {1, 2}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(i^#) = {}, Uargs(c_4) = {},
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
i(x1) = [3] x1 + [3]
j(x1) = [3] x1 + [3]
a() = [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
j^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(c_4) = {}, Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
i(x1) = [0] x1 + [0]
j(x1) = [0] x1 + [0]
a() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [3] x1 + [0]
c_4(x1) = [1] x1 + [0]
j^#(x1) = [0] x1 + [0]
c_5(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: {i^#(x) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(i^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
i^#(x1) = [7] x1 + [7]
c_4(x1) = [1] x1 + [0]
* Path {6}: 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(f) = {}, Uargs(g) = {}, Uargs(h) = {}, Uargs(i) = {},
Uargs(j) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(i^#) = {}, Uargs(c_4) = {}, Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
i(x1) = [0] x1 + [0]
j(x1) = [0] x1 + [0]
a() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
i^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
j^#(x1) = [3] x1 + [0]
c_5(x1) = [1] 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: {j^#(x) -> c_5(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(j^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
j^#(x1) = [7] x1 + [7]
c_5(x1) = [1] x1 + [0]
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.