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:
{ fib(N) -> sel(N, fib1(s(0()), s(0())))
, fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, sel(0(), cons(X, XS)) -> X
, sel(s(N), cons(X, XS)) -> sel(N, XS)}
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: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
, 2: fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
, 3: add^#(0(), X) -> c_2()
, 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
, 5: sel^#(0(), cons(X, XS)) -> c_4()
, 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{5}: NA
-----------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {1}->{6}: inherited
------------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{6}->{5}: NA
----------------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {2}: MAYBE
---------------
The usable rules for this path are:
{ add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel(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]
fib1(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]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
cons(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]
add(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]
fib^#(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]
sel^#(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]
fib1^#(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]
add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [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]
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: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [3]
[0 0 1] [2]
add^#(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
[0 2 1] [0 0 0] [2]
[4 0 2] [0 0 4] [0]
c_3(x1) = [1 0 0] x1 + [1]
[0 0 0] [2]
[2 2 0] [3]
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel(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]
fib1(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
cons(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]
add(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]
fib^#(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]
sel^#(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]
fib1^#(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]
add^#(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_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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: {add^#(0(), X) -> c_2()}
Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [2]
[0 1 4] [0]
[0 0 0] [2]
0() = [2]
[2]
[2]
add^#(x1, x2) = [1 3 1] x1 + [0 0 0] x2 + [0]
[3 2 2] [0 0 4] [0]
[0 2 2] [0 0 2] [2]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 2] [3]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
, 2: fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
, 3: add^#(0(), X) -> c_2()
, 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
, 5: sel^#(0(), cons(X, XS)) -> c_4()
, 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{5}: NA
-----------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {1}->{6}: inherited
------------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{6}->{5}: NA
----------------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {2}: MAYBE
---------------
The usable rules for this path are:
{ add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [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: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_3(x1) = [1 0] x1 + [0]
[0 0] [3]
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(0(), X) -> c_2()}
Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [4]
[0 1] [0]
0() = [2]
[2]
add^#(x1, x2) = [1 3] x1 + [0 0] x2 + [4]
[2 2] [4 4] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
, 2: fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
, 3: add^#(0(), X) -> c_2()
, 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
, 5: sel^#(0(), cons(X, XS)) -> c_4()
, 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{5}: NA
-----------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {1}->{6}: inherited
------------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{6}->{5}: NA
----------------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {2}: MAYBE
---------------
The usable rules for this path are:
{ add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
fib1(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
fib1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
add^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] 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: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
add^#(x1, x2) = [2] x1 + [7] x2 + [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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
fib1(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
fib1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] 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: {add^#(0(), X) -> c_2()}
Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
add^#(x1, x2) = [6] x1 + [7] x2 + [0]
c_2() = [1]
c_3(x1) = [1] x1 + [7]
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:
{ fib(N) -> sel(N, fib1(s(0()), s(0())))
, fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, sel(0(), cons(X, XS)) -> X
, sel(s(N), cons(X, XS)) -> sel(N, XS)}
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: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
, 2: fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
, 3: add^#(0(), X) -> c_2(X)
, 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
, 5: sel^#(0(), cons(X, XS)) -> c_4(X)
, 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{5}: NA
-----------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {1}->{6}: inherited
------------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{6}->{5}: NA
----------------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {2}: MAYBE
---------------
The usable rules for this path are:
{ add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel(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]
fib1(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]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
cons(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]
add(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]
fib^#(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]
sel^#(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]
fib1^#(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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [3]
[0 0 1] [2]
add^#(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
[0 2 1] [0 0 0] [2]
[4 0 2] [0 0 4] [0]
c_3(x1) = [1 0 0] x1 + [1]
[0 0 0] [2]
[2 2 0] [3]
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel(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]
fib1(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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
cons(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]
add(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]
fib^#(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]
sel^#(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]
fib1^#(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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add^#(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_2(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), X) -> c_2(X)}
Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
0() = [2]
[2]
[2]
add^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [0]
[2 2 2] [4 4 4] [0]
[2 2 2] [4 0 4] [0]
c_2(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
, 2: fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
, 3: add^#(0(), X) -> c_2(X)
, 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
, 5: sel^#(0(), cons(X, XS)) -> c_4(X)
, 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{5}: NA
-----------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {1}->{6}: inherited
------------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{6}->{5}: NA
----------------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {2}: MAYBE
---------------
The usable rules for this path are:
{ add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_3(x1) = [1 0] x1 + [0]
[0 0] [3]
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fib1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [1 1] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), X) -> c_2(X)}
Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [2]
[0]
add^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[4 0] [0 4] [0]
c_2(x1) = [0 0] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
, 2: fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
, 3: add^#(0(), X) -> c_2(X)
, 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
, 5: sel^#(0(), cons(X, XS)) -> c_4(X)
, 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ MAYBE ]
->{1} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{5}: NA
-----------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {1}->{6}: inherited
------------------------
This path is subsumed by the proof of path {1}->{6}->{5}.
* Path {1}->{6}->{5}: NA
----------------------
The usable rules for this path are:
{ fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(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.
* Path {2}: MAYBE
---------------
The usable rules for this path are:
{ add(0(), X) -> X
, add(s(X), Y) -> s(add(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:
{ fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))}
Proof Output:
The input cannot be shown compatible
* 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
fib1(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
fib1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
add^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
add^#(x1, x2) = [2] x1 + [7] x2 + [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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
fib(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
fib1(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
fib^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
fib1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
add^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), X) -> c_2(X)}
Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
add^#(x1, x2) = [2] x1 + [7] x2 + [4]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [4]
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.