Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ and(tt(), X) -> X
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))
, x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)}
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: and^#(tt(), X) -> c_0()
, 2: plus^#(N, 0()) -> c_1()
, 3: plus^#(N, s(M)) -> c_2(plus^#(N, M))
, 4: x^#(N, 0()) -> c_3()
, 5: x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_2) = {},
Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
tt() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
x(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]
and^#(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]
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_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
x^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), X) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [2]
[2]
[2]
and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_0() = [0]
[1]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_2) = {},
Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
tt() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
x(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]
and^#(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]
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_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
x^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {x^#(N, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(x^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
x^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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:
{ x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))
, plus^#(N, 0()) -> c_1()
, x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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: and^#(tt(), X) -> c_0()
, 2: plus^#(N, 0()) -> c_1()
, 3: plus^#(N, s(M)) -> c_2(plus^#(N, M))
, 4: x^#(N, 0()) -> c_3()
, 5: x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_2) = {},
Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
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]
x(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
x^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), X) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [2]
[2]
and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_0() = [0]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_2) = {},
Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
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]
x(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
x^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {x^#(N, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(x^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
x^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_3() = [0]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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:
{ x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))
, plus^#(N, 0()) -> c_1()
, x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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: and^#(tt(), X) -> c_0()
, 2: plus^#(N, 0()) -> c_1()
, 3: plus^#(N, s(M)) -> c_2(plus^#(N, M))
, 4: x^#(N, 0()) -> c_3()
, 5: x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_2) = {},
Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
x(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
x^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), X) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [7]
and^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_0() = [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_2) = {},
Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
x(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
x^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {x^#(N, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(x^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
x^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_3() = [1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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:
{ x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))
, plus^#(N, 0()) -> c_1()
, x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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:
{ and(tt(), X) -> X
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))
, x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)}
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: and^#(tt(), X) -> c_0(X)
, 2: plus^#(N, 0()) -> c_1(N)
, 3: plus^#(N, s(M)) -> c_2(plus^#(N, M))
, 4: x^#(N, 0()) -> c_3()
, 5: x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
tt() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
x(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]
and^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 1 1] 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_1(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]
x^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), X) -> c_0(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [2]
[0]
[2]
and^#(x1, x2) = [2 0 2] x1 + [7 7 7] x2 + [7]
[2 0 2] [7 7 7] [7]
[2 0 2] [7 7 7] [7]
c_0(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(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]
tt() = [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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
x(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]
and^#(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]
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_1(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]
x^#(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_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {x^#(N, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(x^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
x^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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:
{ x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))
, plus^#(N, 0()) -> c_1(N)
, x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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: and^#(tt(), X) -> c_0(X)
, 2: plus^#(N, 0()) -> c_1(N)
, 3: plus^#(N, s(M)) -> c_2(plus^#(N, M))
, 4: x^#(N, 0()) -> c_3()
, 5: x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
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]
x(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
x^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), X) -> c_0(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [2]
[2]
and^#(x1, x2) = [2 2] x1 + [7 7] x2 + [7]
[2 2] [7 7] [3]
c_0(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
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]
x(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
x^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {x^#(N, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(x^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
x^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_3() = [0]
[1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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:
{ x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))
, plus^#(N, 0()) -> c_1(N)
, x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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: and^#(tt(), X) -> c_0(X)
, 2: plus^#(N, 0()) -> c_1(N)
, 3: plus^#(N, s(M)) -> c_2(plus^#(N, M))
, 4: x^#(N, 0()) -> c_3()
, 5: x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ inherited ]
|
|->{2} [ MAYBE ]
|
`->{3} [ inherited ]
|
`->{2} [ NA ]
->{4} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
x(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
x^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(tt(), X) -> c_0(X)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
tt() = [5]
and^#(x1, x2) = [3] x1 + [7] x2 + [0]
c_0(x1) = [1] x1 + [0]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(and) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(x) = {},
Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(x^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
and(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
x(x1, x2) = [0] x1 + [0] x2 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
x^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {x^#(N, 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(x^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
x^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_3() = [1]
* Path {5}: inherited
-------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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:
{ x^#(N, s(M)) -> c_4(plus^#(x(N, M), N))
, plus^#(N, 0()) -> c_1(N)
, x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
Proof Output:
The input cannot be shown compatible
* Path {5}->{3}: inherited
------------------------
This path is subsumed by the proof of path {5}->{3}->{2}.
* Path {5}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ x(N, 0()) -> 0()
, x(N, s(M)) -> plus(x(N, M), N)
, plus(N, 0()) -> N
, plus(N, s(M)) -> s(plus(N, M))}
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.