Tool CaT
stdout:
MAYBE
Problem:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
d(0()) -> 0()
d(s(x)) -> s(s(d(x)))
q(0()) -> 0()
q(s(x)) -> s(plus(q(x),d(x)))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(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: plus^#(x, 0()) -> c_0()
, 2: plus^#(x, s(y)) -> c_1(plus^#(x, y))
, 3: d^#(0()) -> c_2()
, 4: d^#(s(x)) -> c_3(d^#(x))
, 5: q^#(0()) -> c_4()
, 6: q^#(s(x)) -> c_5(plus^#(q(x), d(x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {4}: 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(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
d(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
q(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
q^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [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(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {d^#(s(x)) -> c_3(d^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
d^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_3(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {4}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
q(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
q^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [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(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {d^#(0()) -> c_2()}
Weak Rules: {d^#(s(x)) -> c_3(d^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
d^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* 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(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
q(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
q^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {q^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(q^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
q^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_4() = [0]
[1]
[1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ q^#(s(x)) -> c_5(plus^#(q(x), d(x)))
, plus^#(x, 0()) -> c_0()
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: inherited
------------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, 0()) -> c_0()
, 2: plus^#(x, s(y)) -> c_1(plus^#(x, y))
, 3: d^#(0()) -> c_2()
, 4: d^#(s(x)) -> c_3(d^#(x))
, 5: q^#(0()) -> c_4()
, 6: q^#(s(x)) -> c_5(plus^#(q(x), d(x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
q(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
q^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [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(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {d^#(s(x)) -> c_3(d^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
d^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
q(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
q^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [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(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {d^#(0()) -> c_2()}
Weak Rules: {d^#(s(x)) -> c_3(d^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
d^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [5]
[2 0] [3]
* 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(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
q(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
q^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {q^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(q^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
q^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_4() = [0]
[1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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:
{ q^#(s(x)) -> c_5(plus^#(q(x), d(x)))
, plus^#(x, 0()) -> c_0()
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: inherited
------------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, 0()) -> c_0()
, 2: plus^#(x, s(y)) -> c_1(plus^#(x, y))
, 3: d^#(0()) -> c_2()
, 4: d^#(s(x)) -> c_3(d^#(x))
, 5: q^#(0()) -> c_4()
, 6: q^#(s(x)) -> c_5(plus^#(q(x), d(x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
q(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [3] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
q^#(x1) = [0] x1 + [0]
c_4() = [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(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {d^#(s(x)) -> c_3(d^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
d^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
q(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
q^#(x1) = [0] x1 + [0]
c_4() = [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(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {d^#(0()) -> c_2()}
Weak Rules: {d^#(s(x)) -> c_3(d^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
d^#(x1) = [2] x1 + [0]
c_2() = [1]
c_3(x1) = [1] x1 + [0]
* 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(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(d^#) = {},
Uargs(c_3) = {}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
q(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
q^#(x1) = [0] x1 + [0]
c_4() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {q^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(q^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
q^#(x1) = [1] x1 + [7]
c_4() = [1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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:
{ q^#(s(x)) -> c_5(plus^#(q(x), d(x)))
, plus^#(x, 0()) -> c_0()
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: inherited
------------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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.
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:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(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: plus^#(x, 0()) -> c_0(x)
, 2: plus^#(x, s(y)) -> c_1(plus^#(x, y))
, 3: d^#(0()) -> c_2()
, 4: d^#(s(x)) -> c_3(d^#(x))
, 5: q^#(0()) -> c_4()
, 6: q^#(s(x)) -> c_5(plus^#(q(x), d(x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {4}: 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(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
d(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
q(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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]
d^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
q^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [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(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {d^#(s(x)) -> c_3(d^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
d^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_3(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {4}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
q(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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]
d^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
q^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [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(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {d^#(0()) -> c_2()}
Weak Rules: {d^#(s(x)) -> c_3(d^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
d^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* 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(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
q(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus^#(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]
d^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
q^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [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: {q^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(q^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
q^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_4() = [0]
[1]
[1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ q^#(s(x)) -> c_5(plus^#(q(x), d(x)))
, plus^#(x, 0()) -> c_0(x)
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: inherited
------------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, 0()) -> c_0(x)
, 2: plus^#(x, s(y)) -> c_1(plus^#(x, y))
, 3: d^#(0()) -> c_2()
, 4: d^#(s(x)) -> c_3(d^#(x))
, 5: q^#(0()) -> c_4()
, 6: q^#(s(x)) -> c_5(plus^#(q(x), d(x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
q(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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]
d^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
q^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [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(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {d^#(s(x)) -> c_3(d^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
d^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
q(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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]
d^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
q^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [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(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {d^#(0()) -> c_2()}
Weak Rules: {d^#(s(x)) -> c_3(d^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
d^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [5]
[2 0] [3]
* 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(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1) = [0 0] x1 + [0]
[0 0] [0]
q(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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]
d^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
q^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [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: {q^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(q^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
q^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_4() = [0]
[1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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:
{ q^#(s(x)) -> c_5(plus^#(q(x), d(x)))
, plus^#(x, 0()) -> c_0(x)
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: inherited
------------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, 0()) -> c_0(x)
, 2: plus^#(x, s(y)) -> c_1(plus^#(x, y))
, 3: d^#(0()) -> c_2()
, 4: d^#(s(x)) -> c_3(d^#(x))
, 5: q^#(0()) -> c_4()
, 6: q^#(s(x)) -> c_5(plus^#(q(x), d(x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
q(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [3] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
q^#(x1) = [0] x1 + [0]
c_4() = [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(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {d^#(s(x)) -> c_3(d^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
d^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {1}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
q(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
q^#(x1) = [0] x1 + [0]
c_4() = [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(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {d^#(0()) -> c_2()}
Weak Rules: {d^#(s(x)) -> c_3(d^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(d^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
d^#(x1) = [2] x1 + [0]
c_2() = [1]
c_3(x1) = [1] x1 + [0]
* 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(plus) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(q) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(d^#) = {}, Uargs(c_3) = {}, Uargs(q^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
q(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
q^#(x1) = [0] x1 + [0]
c_4() = [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: {q^#(0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(q^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
q^#(x1) = [1] x1 + [7]
c_4() = [1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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:
{ q^#(s(x)) -> c_5(plus^#(q(x), d(x)))
, plus^#(x, 0()) -> c_0(x)
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: inherited
------------------------
This path is subsumed by the proof of path {6}->{2}->{1}.
* Path {6}->{2}->{1}: NA
----------------------
The usable rules for this path are:
{ d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))
, plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))}
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.
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 pair1rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3rc
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
stdout:
MAYBE
We consider the following Problem:
Strict Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: none
Certificate: MAYBE
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Sequentially' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'matrix-interpretation of dimension 4 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
2) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
3) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Bounds with minimal-enrichment and initial automaton 'match' (timeout of 100.0 seconds)' failed due to the following reason:
match-boundness of the problem could not be verified.
2) 'Bounds with perSymbol-enrichment and initial automaton 'match' (timeout of 5.0 seconds)' failed due to the following reason:
match-boundness of the problem could not be verified.
2) 'dp' failed due to the following reason:
We have computed the following dependency pairs
Strict Dependency Pairs:
{ plus^#(x, 0()) -> c_1(x)
, plus^#(x, s(y)) -> c_2(plus^#(x, y))
, d^#(0()) -> c_3()
, d^#(s(x)) -> c_4(d^#(x))
, q^#(0()) -> c_5()
, q^#(s(x)) -> c_6(plus^#(q(x), d(x)))}
We consider the following Problem:
Strict DPs:
{ plus^#(x, 0()) -> c_1(x)
, plus^#(x, s(y)) -> c_2(plus^#(x, y))
, d^#(0()) -> c_3()
, d^#(s(x)) -> c_4(d^#(x))
, q^#(0()) -> c_5()
, q^#(s(x)) -> c_6(plus^#(q(x), d(x)))}
Strict Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: none
Certificate: MAYBE
Application of 'usablerules':
-----------------------------
All rules are usable.
No subproblems were generated.
Arrrr..Tool tup3irc
| Execution Time | 44.136806ms |
|---|
| Answer | MAYBE |
|---|
| Input | TCT 09 ma4 |
|---|
stdout:
MAYBE
We consider the following Problem:
Strict Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: innermost
Certificate: MAYBE
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
The input problem contains no overlaps that give rise to inapplicable rules.
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: innermost
1) None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'dp' failed due to the following reason:
We have computed the following dependency pairs
Strict Dependency Pairs:
{ plus^#(x, 0()) -> c_1()
, plus^#(x, s(y)) -> c_2(plus^#(x, y))
, d^#(0()) -> c_3()
, d^#(s(x)) -> c_4(d^#(x))
, q^#(0()) -> c_5()
, q^#(s(x)) -> c_6(plus^#(q(x), d(x)), q^#(x), d^#(x))}
We consider the following Problem:
Strict DPs:
{ plus^#(x, 0()) -> c_1()
, plus^#(x, s(y)) -> c_2(plus^#(x, y))
, d^#(0()) -> c_3()
, d^#(s(x)) -> c_4(d^#(x))
, q^#(0()) -> c_5()
, q^#(s(x)) -> c_6(plus^#(q(x), d(x)), q^#(x), d^#(x))}
Weak Trs:
{ plus(x, 0()) -> x
, plus(x, s(y)) -> s(plus(x, y))
, d(0()) -> 0()
, d(s(x)) -> s(s(d(x)))
, q(0()) -> 0()
, q(s(x)) -> s(plus(q(x), d(x)))}
StartTerms: basic terms
Strategy: innermost
Certificate: MAYBE
Application of 'Fastest':
-------------------------
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
'Sequentially' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
2) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Sequentially' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'matrix-interpretation of dimension 4 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
2) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
3) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
The input cannot be shown compatible
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Arrrr..