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