Tool CaT
stdout:
MAYBE
Problem:
f(s(x)) -> f(id_inc(c(x,x)))
f(c(s(x),y)) -> g(c(x,y))
g(c(s(x),y)) -> g(c(y,x))
g(c(x,s(y))) -> g(c(y,x))
g(c(x,x)) -> f(x)
id_inc(c(x,y)) -> c(id_inc(x),id_inc(y))
id_inc(s(x)) -> s(id_inc(x))
id_inc(0()) -> 0()
id_inc(0()) -> s(0())
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:
{ f(s(x)) -> f(id_inc(c(x, x)))
, f(c(s(x), y)) -> g(c(x, y))
, g(c(s(x), y)) -> g(c(y, x))
, g(c(x, s(y))) -> g(c(y, x))
, g(c(x, x)) -> f(x)
, id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
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^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, 2: f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, 3: g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, 4: g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, 5: g^#(c(x, x)) -> c_4(f^#(x))
, 6: id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, 7: id_inc^#(s(x)) -> c_6(id_inc^#(x))
, 8: id_inc^#(0()) -> c_7()
, 9: id_inc^#(0()) -> c_8()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ YES(?,O(n^3)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{1,5,4,3,2} [ NA ]
Sub-problems:
-------------
* Path {1,5,4,3,2}: NA
--------------------
The usable rules for this path are:
{ id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6,7}: YES(?,O(n^3))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
id_inc(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(x1, x2) = [1 3 0] x1 + [1 3 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
id_inc^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0 2] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
c(x1, x2) = [1 5 6] x1 + [1 4 6] x2 + [0]
[0 1 3] [0 0 1] [0]
[0 0 1] [0 0 1] [4]
id_inc^#(x1) = [1 0 2] x1 + [0]
[0 2 2] [2]
[0 0 0] [3]
c_5(x1, x2) = [1 0 0] x1 + [1 2 0] x2 + [3]
[0 1 2] [0 0 0] [1]
[0 0 0] [0 0 0] [3]
c_6(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [3]
* Path {6,7}->{8}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
id_inc(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(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]
0() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
id_inc^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_7() = [0]
[0]
[0]
c_8() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {id_inc^#(0()) -> c_7()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(x1, x2) = [1 4 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [2]
[0]
[0]
id_inc^#(x1) = [2 0 0] x1 + [0]
[1 0 0] [0]
[0 0 0] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [1]
[0]
[0]
* Path {6,7}->{9}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
id_inc(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(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]
0() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
id_inc^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_7() = [0]
[0]
[0]
c_8() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {id_inc^#(0()) -> c_8()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(x1, x2) = [1 4 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [2]
[0]
[0]
id_inc^#(x1) = [2 0 0] x1 + [0]
[1 0 0] [0]
[0 0 0] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [1]
[0]
[0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, 2: f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, 3: g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, 4: g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, 5: g^#(c(x, x)) -> c_4(f^#(x))
, 6: id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, 7: id_inc^#(s(x)) -> c_6(id_inc^#(x))
, 8: id_inc^#(0()) -> c_7()
, 9: id_inc^#(0()) -> c_8()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ YES(?,O(n^2)) ]
|
|->{8} [ YES(?,O(n^2)) ]
|
`->{9} [ YES(?,O(n^2)) ]
->{1,5,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,5,4,3,2}: MAYBE
-----------------------
The usable rules for this path are:
{ id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, g^#(c(x, x)) -> c_4(f^#(x))
, g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
Proof Output:
The input cannot be shown compatible
* Path {6,7}: YES(?,O(n^2))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
id_inc(x1) = [0 0] x1 + [0]
[0 0] [0]
c(x1, x2) = [1 1] x1 + [1 2] x2 + [0]
[0 1] [0 1] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
c(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [2]
id_inc^#(x1) = [0 4] x1 + [1]
[0 0] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [6]
[0 0] [0 0] [0]
c_6(x1) = [1 0] x1 + [5]
[0 0] [0]
* Path {6,7}->{8}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc(x1) = [0 0] x1 + [0]
[0 0] [0]
c(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {id_inc^#(0()) -> c_7()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3] x1 + [3]
[0 0] [0]
c(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
[0 1] [0 1] [3]
0() = [2]
[0]
id_inc^#(x1) = [3 2] x1 + [0]
[3 3] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [7]
[0 0] [0 0] [6]
c_6(x1) = [1 0] x1 + [7]
[0 0] [7]
c_7() = [1]
[0]
* Path {6,7}->{9}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc(x1) = [0 0] x1 + [0]
[0 0] [0]
c(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {id_inc^#(0()) -> c_8()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3] x1 + [3]
[0 0] [0]
c(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
[0 1] [0 1] [3]
0() = [2]
[0]
id_inc^#(x1) = [3 2] x1 + [0]
[3 3] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [7]
[0 0] [0 0] [6]
c_6(x1) = [1 0] x1 + [7]
[0 0] [7]
c_8() = [1]
[0]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, 2: f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, 3: g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, 4: g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, 5: g^#(c(x, x)) -> c_4(f^#(x))
, 6: id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, 7: id_inc^#(s(x)) -> c_6(id_inc^#(x))
, 8: id_inc^#(0()) -> c_7()
, 9: id_inc^#(0()) -> c_8()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{1,5,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,5,4,3,2}: MAYBE
-----------------------
The usable rules for this path are:
{ id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, g^#(c(x, x)) -> c_4(f^#(x))
, g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
Proof Output:
The input cannot be shown compatible
* Path {6,7}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
id_inc(x1) = [0] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
id_inc^#(x1) = [3] x1 + [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
c(x1, x2) = [1] x1 + [1] x2 + [4]
id_inc^#(x1) = [2] x1 + [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [7]
c_6(x1) = [1] x1 + [7]
* Path {6,7}->{8}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
id_inc(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
id_inc^#(x1) = [0] x1 + [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {id_inc^#(0()) -> c_7()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [2]
id_inc^#(x1) = [3] x1 + [2]
c_5(x1, x2) = [1] x1 + [1] x2 + [3]
c_6(x1) = [1] x1 + [0]
c_7() = [1]
* Path {6,7}->{9}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
id_inc(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
id_inc^#(x1) = [0] x1 + [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {id_inc^#(0()) -> c_8()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [2]
id_inc^#(x1) = [3] x1 + [2]
c_5(x1, x2) = [1] x1 + [1] x2 + [3]
c_6(x1) = [1] x1 + [0]
c_8() = [1]
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:
{ f(s(x)) -> f(id_inc(c(x, x)))
, f(c(s(x), y)) -> g(c(x, y))
, g(c(s(x), y)) -> g(c(y, x))
, g(c(x, s(y))) -> g(c(y, x))
, g(c(x, x)) -> f(x)
, id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
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^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, 2: f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, 3: g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, 4: g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, 5: g^#(c(x, x)) -> c_4(f^#(x))
, 6: id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, 7: id_inc^#(s(x)) -> c_6(id_inc^#(x))
, 8: id_inc^#(0()) -> c_7()
, 9: id_inc^#(0()) -> c_8()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ YES(?,O(n^3)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{1,5,4,3,2} [ NA ]
Sub-problems:
-------------
* Path {1,5,4,3,2}: NA
--------------------
The usable rules for this path are:
{ id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6,7}: YES(?,O(n^3))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
id_inc(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(x1, x2) = [1 3 0] x1 + [1 3 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
id_inc^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0 2] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
c(x1, x2) = [1 5 6] x1 + [1 4 6] x2 + [0]
[0 1 3] [0 0 1] [0]
[0 0 1] [0 0 1] [4]
id_inc^#(x1) = [1 0 2] x1 + [0]
[0 2 2] [2]
[0 0 0] [3]
c_5(x1, x2) = [1 0 0] x1 + [1 2 0] x2 + [3]
[0 1 2] [0 0 0] [1]
[0 0 0] [0 0 0] [3]
c_6(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [3]
* Path {6,7}->{8}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
id_inc(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(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]
0() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
id_inc^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_7() = [0]
[0]
[0]
c_8() = [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: {id_inc^#(0()) -> c_7()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(x1, x2) = [1 4 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [2]
[0]
[0]
id_inc^#(x1) = [2 0 0] x1 + [0]
[1 0 0] [0]
[0 0 0] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [1]
[0]
[0]
* Path {6,7}->{9}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
id_inc(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(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]
0() = [0]
[0]
[0]
f^#(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]
c_1(x1) = [0 0 0] x1 + [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_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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]
id_inc^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_7() = [0]
[0]
[0]
c_8() = [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: {id_inc^#(0()) -> c_8()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c(x1, x2) = [1 4 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [2]
[0]
[0]
id_inc^#(x1) = [2 0 0] x1 + [0]
[1 0 0] [0]
[0 0 0] [0]
c_5(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [1]
[0]
[0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, 2: f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, 3: g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, 4: g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, 5: g^#(c(x, x)) -> c_4(f^#(x))
, 6: id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, 7: id_inc^#(s(x)) -> c_6(id_inc^#(x))
, 8: id_inc^#(0()) -> c_7()
, 9: id_inc^#(0()) -> c_8()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ YES(?,O(n^2)) ]
|
|->{8} [ YES(?,O(n^2)) ]
|
`->{9} [ YES(?,O(n^2)) ]
->{1,5,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,5,4,3,2}: MAYBE
-----------------------
The usable rules for this path are:
{ id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, g^#(c(x, x)) -> c_4(f^#(x))
, g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
Proof Output:
The input cannot be shown compatible
* Path {6,7}: YES(?,O(n^2))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
id_inc(x1) = [0 0] x1 + [0]
[0 0] [0]
c(x1, x2) = [1 1] x1 + [1 2] x2 + [0]
[0 1] [0 1] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8() = [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:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
c(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [2]
id_inc^#(x1) = [0 4] x1 + [1]
[0 0] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [6]
[0 0] [0 0] [0]
c_6(x1) = [1 0] x1 + [5]
[0 0] [0]
* Path {6,7}->{8}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc(x1) = [0 0] x1 + [0]
[0 0] [0]
c(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {id_inc^#(0()) -> c_7()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3] x1 + [3]
[0 0] [0]
c(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
[0 1] [0 1] [3]
0() = [2]
[0]
id_inc^#(x1) = [3 2] x1 + [0]
[3 3] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [7]
[0 0] [0 0] [6]
c_6(x1) = [1 0] x1 + [7]
[0 0] [7]
c_7() = [1]
[0]
* Path {6,7}->{9}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc(x1) = [0 0] x1 + [0]
[0 0] [0]
c(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
id_inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {id_inc^#(0()) -> c_8()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 3] x1 + [3]
[0 0] [0]
c(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
[0 1] [0 1] [3]
0() = [2]
[0]
id_inc^#(x1) = [3 2] x1 + [0]
[3 3] [0]
c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [7]
[0 0] [0 0] [6]
c_6(x1) = [1 0] x1 + [7]
[0 0] [7]
c_8() = [1]
[0]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, 2: f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, 3: g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, 4: g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, 5: g^#(c(x, x)) -> c_4(f^#(x))
, 6: id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, 7: id_inc^#(s(x)) -> c_6(id_inc^#(x))
, 8: id_inc^#(0()) -> c_7()
, 9: id_inc^#(0()) -> c_8()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{1,5,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,5,4,3,2}: MAYBE
-----------------------
The usable rules for this path are:
{ id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f^#(s(x)) -> c_0(f^#(id_inc(c(x, x))))
, g^#(c(x, x)) -> c_4(f^#(x))
, g^#(c(x, s(y))) -> c_3(g^#(c(y, x)))
, g^#(c(s(x), y)) -> c_2(g^#(c(y, x)))
, f^#(c(s(x), y)) -> c_1(g^#(c(x, y)))
, id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
, id_inc(s(x)) -> s(id_inc(x))
, id_inc(0()) -> 0()
, id_inc(0()) -> s(0())}
Proof Output:
The input cannot be shown compatible
* Path {6,7}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
id_inc(x1) = [0] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
id_inc^#(x1) = [3] x1 + [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [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:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
c(x1, x2) = [1] x1 + [1] x2 + [4]
id_inc^#(x1) = [2] x1 + [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [7]
c_6(x1) = [1] x1 + [7]
* Path {6,7}->{8}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
id_inc(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
id_inc^#(x1) = [0] x1 + [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [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: {id_inc^#(0()) -> c_7()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [2]
id_inc^#(x1) = [3] x1 + [2]
c_5(x1, x2) = [1] x1 + [1] x2 + [3]
c_6(x1) = [1] x1 + [0]
c_7() = [1]
* Path {6,7}->{9}: 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(f) = {}, Uargs(s) = {}, Uargs(id_inc) = {}, Uargs(c) = {},
Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(id_inc^#) = {}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
id_inc(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
0() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
id_inc^#(x1) = [0] x1 + [0]
c_5(x1, x2) = [1] x1 + [1] x2 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8() = [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: {id_inc^#(0()) -> c_8()}
Weak Rules:
{ id_inc^#(c(x, y)) -> c_5(id_inc^#(x), id_inc^#(y))
, id_inc^#(s(x)) -> c_6(id_inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(c) = {}, Uargs(id_inc^#) = {},
Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [2]
id_inc^#(x1) = [3] x1 + [2]
c_5(x1, x2) = [1] x1 + [1] x2 + [3]
c_6(x1) = [1] x1 + [0]
c_8() = [1]
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.