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:
{ natsFrom(N) -> cons(N, natsFrom(s(N)))
, fst(pair(XS, YS)) -> XS
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
, head(cons(N, XS)) -> N
, tail(cons(N, XS)) -> XS
, sel(N, XS) -> head(afterNth(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, afterNth(N, XS) -> snd(splitAt(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: natsFrom^#(N) -> c_0(natsFrom^#(s(N)))
, 2: fst^#(pair(XS, YS)) -> c_1()
, 3: snd^#(pair(XS, YS)) -> c_2()
, 4: splitAt^#(0(), XS) -> c_3()
, 5: splitAt^#(s(N), cons(X, XS)) ->
c_4(u^#(splitAt(N, XS), N, X, XS))
, 6: u^#(pair(YS, ZS), N, X, XS) -> c_5()
, 7: head^#(cons(N, XS)) -> c_6()
, 8: tail^#(cons(N, XS)) -> c_7()
, 9: sel^#(N, XS) -> c_8(head^#(afterNth(N, XS)))
, 10: take^#(N, XS) -> c_9(fst^#(splitAt(N, XS)))
, 11: afterNth^#(N, XS) -> c_10(snd^#(splitAt(N, XS)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ NA ]
|
`->{3} [ NA ]
->{10} [ NA ]
|
`->{2} [ NA ]
->{9} [ NA ]
|
`->{7} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(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(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {1},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(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]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(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]
u(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 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]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(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: {natsFrom^#(N) -> c_0(natsFrom^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(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]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(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]
u(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 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]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {splitAt^#(0(), XS) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(splitAt^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
splitAt^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_3() = [0]
[1]
[1]
* Path {5}: NA
------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
[0 0 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [1]
[0 1 0] [2]
[0 0 0] [1]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 1] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 1 0] x1 + [3 1 0] x2 + [1]
[3 0 3] [0 0 0] [2]
[0 0 0] [3 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [1 0 1] x3 + [0 0 0] x4 + [1]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(x1, x2) = [0 1 3] x1 + [3 0 3] 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]
u^#(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [1 0 1] x3 + [0 0 0] x4 + [0]
[3 3 3] [3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [3 3 3] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {5}->{6}: NA
-----------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 0] [3]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 0 0] x1 + [0 1 0] x2 + [2]
[0 1 0] [0 0 0] [0]
[0 1 0] [1 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [2]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(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]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(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]
u(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 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]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(N, XS)) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
tail^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_7() = [0]
[1]
[1]
* Path {9}: NA
------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {1},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 2] x1 + [1 2 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 1] [0]
s(x1) = [1 0 1] x1 + [1]
[0 1 0] [0]
[0 0 1] [1]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [1 0 1] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [1 0 0] x1 + [2]
[0 1 0] [3]
[2 0 0] [3]
splitAt(x1, x2) = [1 0 0] x1 + [1 0 1] x2 + [0]
[3 3 1] [1 2 0] [0]
[0 0 1] [0 0 0] [0]
0() = [3]
[2]
[1]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 1] x1 + [0 0 0] x2 + [1 0 1] x3 + [0 0 0] x4 + [0]
[0 1 0] [0 0 0] [1 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(x1, x2) = [1 0 0] x1 + [1 0 1] x2 + [3]
[3 3 2] [2 2 0] [3]
[2 2 2] [2 2 2] [3]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [2 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
sel^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {9}->{7}: NA
-----------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {1},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 2 1] x2 + [1]
[0 0 0] [0 0 3] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [3]
snd(x1) = [1 0 0] x1 + [2]
[1 0 0] [3]
[1 0 0] [3]
splitAt(x1, x2) = [1 2 0] x1 + [2 2 2] x2 + [0]
[0 0 0] [0 3 0] [0]
[0 1 0] [0 0 0] [0]
0() = [0]
[3]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 3] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 2] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 1] [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]
tail(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]
afterNth(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [3]
[3 3 3] [3 3 3] [3]
[3 3 3] [3 3 3] [3]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {10}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [3]
[0 1 2] [2]
[0 0 1] [3]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [1]
[3 1 1] [0 0 0] [2]
[2 0 0] [1 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [2]
[3 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [2]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [3 0 1] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_9(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{2}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 0] [3]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 0 0] x1 + [0 1 0] x2 + [2]
[0 1 0] [0 0 0] [0]
[0 1 0] [1 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [2]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {11}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {1}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [3]
[0 1 2] [2]
[0 0 1] [3]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [1]
[3 1 1] [0 0 0] [2]
[2 0 0] [1 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [2]
[3 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [2]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [3 0 1] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {11}->{3}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {1}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 0] [3]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 0 0] x1 + [0 1 0] x2 + [2]
[0 1 0] [0 0 0] [0]
[0 1 0] [1 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [2]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
snd^#(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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: natsFrom^#(N) -> c_0(natsFrom^#(s(N)))
, 2: fst^#(pair(XS, YS)) -> c_1()
, 3: snd^#(pair(XS, YS)) -> c_2()
, 4: splitAt^#(0(), XS) -> c_3()
, 5: splitAt^#(s(N), cons(X, XS)) ->
c_4(u^#(splitAt(N, XS), N, X, XS))
, 6: u^#(pair(YS, ZS), N, X, XS) -> c_5()
, 7: head^#(cons(N, XS)) -> c_6()
, 8: tail^#(cons(N, XS)) -> c_7()
, 9: sel^#(N, XS) -> c_8(head^#(afterNth(N, XS)))
, 10: take^#(N, XS) -> c_9(fst^#(splitAt(N, XS)))
, 11: afterNth^#(N, XS) -> c_10(snd^#(splitAt(N, XS)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ NA ]
|
`->{3} [ NA ]
->{10} [ NA ]
|
`->{2} [ NA ]
->{9} [ NA ]
|
`->{7} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(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(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {1},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
u(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(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: {natsFrom^#(N) -> c_0(natsFrom^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
u(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {splitAt^#(0(), XS) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(splitAt^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
splitAt^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_3() = [0]
[1]
* Path {5}: NA
------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [1 3] x2 + [1]
[0 1] [0 0] [0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [1 0] x1 + [3 0] x2 + [2]
[0 0] [3 2] [2]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [2]
[0 0] [0 0] [1 3] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(x1, x2) = [1 3] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
u^#(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [0]
[3 3] [3 3] [3 3] [3 3] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {5}->{6}: NA
-----------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [1]
[0 1] [1]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [2 1] x1 + [0 0] x2 + [1]
[0 0] [2 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [2]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
u(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(N, XS)) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
tail^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_7() = [0]
[1]
* Path {9}: NA
------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {1},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 3] x2 + [3]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 1] x2 + [1]
[0 0] [0 1] [0]
snd(x1) = [1 0] x1 + [0]
[0 1] [3]
splitAt(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [3 1] [0]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [2]
[2 0] [0 0] [1 0] [0 2] [2]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [1 1] x2 + [3]
[0 2] [3 2] [3]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [2 0] x1 + [0]
[3 3] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {9}->{7}: NA
-----------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {1},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 3] x2 + [2]
[0 0] [0 1] [1]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 1] x2 + [1]
[0 0] [0 1] [1]
snd(x1) = [1 0] x1 + [0]
[0 1] [1]
splitAt(x1, x2) = [0 1] x1 + [1 2] x2 + [2]
[0 1] [1 1] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [2]
[1 0] [0 0] [1 0] [0 2] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [3 3] x1 + [3 3] x2 + [3]
[3 3] [3 3] [3]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {10}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [3]
[0 0] [0 1] [3]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 1] x1 + [3 3] x2 + [2]
[0 1] [0 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [2]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [1 1] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{2}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [1]
[0 1] [1]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [2 1] x1 + [0 0] x2 + [1]
[0 0] [2 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [2]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {11}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {1}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [3]
[0 0] [0 1] [3]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 1] x1 + [3 3] x2 + [2]
[0 1] [0 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [2]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [1 1] x1 + [0]
[3 3] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {11}->{3}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {1}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [1]
[0 1] [1]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [2 1] x1 + [0 0] x2 + [1]
[0 0] [2 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [2]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
snd^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
splitAt^#(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]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: natsFrom^#(N) -> c_0(natsFrom^#(s(N)))
, 2: fst^#(pair(XS, YS)) -> c_1()
, 3: snd^#(pair(XS, YS)) -> c_2()
, 4: splitAt^#(0(), XS) -> c_3()
, 5: splitAt^#(s(N), cons(X, XS)) ->
c_4(u^#(splitAt(N, XS), N, X, XS))
, 6: u^#(pair(YS, ZS), N, X, XS) -> c_5()
, 7: head^#(cons(N, XS)) -> c_6()
, 8: tail^#(cons(N, XS)) -> c_7()
, 9: sel^#(N, XS) -> c_8(head^#(afterNth(N, XS)))
, 10: take^#(N, XS) -> c_9(fst^#(splitAt(N, XS)))
, 11: afterNth^#(N, XS) -> c_10(snd^#(splitAt(N, XS)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ NA ]
|
`->{3} [ NA ]
->{10} [ NA ]
|
`->{2} [ NA ]
->{9} [ NA ]
|
`->{7} [ NA ]
->{8} [ YES(?,O(1)) ]
->{5} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(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(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {1},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
u(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(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: {natsFrom^#(N) -> c_0(natsFrom^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
u(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {splitAt^#(0(), XS) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(splitAt^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
splitAt^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_3() = [1]
* Path {5}: NA
------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [2] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [0] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
u^#(x1, x2, x3, x4) = [1] x1 + [3] x2 + [1] x3 + [1] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5}->{6}: NA
-----------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [3] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
u^#(x1, x2, x3, x4) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
u(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(N, XS)) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_7() = [1]
* Path {9}: NA
------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {1},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [1]
snd(x1) = [1] x1 + [0]
splitAt(x1, x2) = [0] x1 + [3] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [3] x1 + [3] x2 + [3]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [1] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_8(x1) = [1] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {9}->{7}: NA
-----------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {1},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [1] x1 + [1]
splitAt(x1, x2) = [0] x1 + [2] x2 + [1]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [0] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [3] x1 + [3] x2 + [3]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [3] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [1] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [3] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_9(x1) = [1] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{2}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(snd^#) = {}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [3] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [3] x1 + [0]
c_1() = [0]
snd^#(x1) = [0] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [1] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {1}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [1] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [3] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_10(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}->{3}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(snd^#) = {1}, Uargs(splitAt^#) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(head^#) = {},
Uargs(tail^#) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [3] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
fst^#(x1) = [0] x1 + [0]
c_1() = [0]
snd^#(x1) = [3] x1 + [0]
c_2() = [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5() = [0]
head^#(x1) = [0] x1 + [0]
c_6() = [0]
tail^#(x1) = [0] x1 + [0]
c_7() = [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ natsFrom(N) -> cons(N, natsFrom(s(N)))
, fst(pair(XS, YS)) -> XS
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
, head(cons(N, XS)) -> N
, tail(cons(N, XS)) -> XS
, sel(N, XS) -> head(afterNth(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, afterNth(N, XS) -> snd(splitAt(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: natsFrom^#(N) -> c_0(N, natsFrom^#(s(N)))
, 2: fst^#(pair(XS, YS)) -> c_1(XS)
, 3: snd^#(pair(XS, YS)) -> c_2(YS)
, 4: splitAt^#(0(), XS) -> c_3(XS)
, 5: splitAt^#(s(N), cons(X, XS)) ->
c_4(u^#(splitAt(N, XS), N, X, XS))
, 6: u^#(pair(YS, ZS), N, X, XS) -> c_5(X, YS, ZS)
, 7: head^#(cons(N, XS)) -> c_6(N)
, 8: tail^#(cons(N, XS)) -> c_7(XS)
, 9: sel^#(N, XS) -> c_8(head^#(afterNth(N, XS)))
, 10: take^#(N, XS) -> c_9(fst^#(splitAt(N, XS)))
, 11: afterNth^#(N, XS) -> c_10(snd^#(splitAt(N, XS)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ NA ]
|
`->{3} [ NA ]
->{10} [ NA ]
|
`->{2} [ NA ]
->{9} [ NA ]
|
`->{7} [ NA ]
->{8} [ YES(?,O(n^3)) ]
->{5} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(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(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {2},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(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]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(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]
u(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 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]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(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]
snd^#(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]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(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: {natsFrom^#(N) -> c_0(N, natsFrom^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(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]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(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]
u(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 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]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(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]
snd^#(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]
splitAt^#(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_3(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {splitAt^#(0(), XS) -> c_3(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(splitAt^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[2]
splitAt^#(x1, x2) = [2 0 2] x1 + [7 7 7] x2 + [7]
[2 0 2] [7 7 7] [7]
[2 0 2] [7 7 7] [7]
c_3(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {5}: NA
------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
[0 0 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [1]
[0 1 0] [2]
[0 0 0] [1]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 1] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 1 0] x1 + [3 1 0] x2 + [1]
[3 0 3] [0 0 0] [2]
[0 0 0] [3 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [1 0 1] x3 + [0 0 0] x4 + [1]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(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]
snd^#(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]
splitAt^#(x1, x2) = [0 1 3] x1 + [3 0 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
u^#(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [1 0 1] x3 + [0 0 0] x4 + [0]
[3 3 3] [3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [3 3 3] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {5}->{6}: NA
-----------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [0]
[0 0 0] [0]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(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]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [2 0 0] x1 + [0 1 0] x2 + [1]
[0 1 0] [0 0 0] [0]
[0 1 0] [0 1 0] [0]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [2]
[0 0 0] [0 0 0] [0 0 0] [0 0 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]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(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]
snd^#(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]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
u^#(x1, x2, x3, x4) = [3 3 3] x1 + [3 3 3] x2 + [0 0 0] x3 + [3 3 3] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [1 1 1] x1 + [1 1 1] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {8}: 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(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
[0 1 1] [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]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(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]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(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]
u(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 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]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(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]
snd^#(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]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [1 0 1] 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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(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: {tail^#(cons(N, XS)) -> c_7(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_7) = {}
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 2] [2]
[0 0 0] [0 0 0] [2]
tail^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {9}: NA
------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {1}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 2] x1 + [1 2 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 1] [0]
s(x1) = [1 0 1] x1 + [1]
[0 1 0] [0]
[0 0 1] [1]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 0] x1 + [1 0 1] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [1 0 0] x1 + [2]
[0 1 0] [3]
[2 0 0] [3]
splitAt(x1, x2) = [1 0 0] x1 + [1 0 1] x2 + [0]
[3 3 1] [1 2 0] [0]
[0 0 1] [0 0 0] [0]
0() = [3]
[2]
[1]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 1] x1 + [0 0 0] x2 + [1 0 1] x3 + [0 0 0] x4 + [0]
[0 1 0] [0 0 0] [1 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(x1, x2) = [1 0 0] x1 + [1 0 1] x2 + [3]
[3 3 2] [2 2 0] [3]
[2 2 2] [2 2 2] [3]
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]
natsFrom^#(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]
fst^#(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]
snd^#(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]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
head^#(x1) = [2 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sel^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {9}->{7}: NA
-----------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {1}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 2 1] x1 + [1 3 2] x2 + [0]
[0 1 1] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 3 3] x1 + [0]
[0 0 0] [3]
[0 0 0] [3]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(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]
snd(x1) = [1 0 0] x1 + [2]
[0 1 0] [3]
[0 0 2] [3]
splitAt(x1, x2) = [1 3 3] x1 + [1 0 0] x2 + [0]
[0 0 0] [3 1 0] [0]
[0 0 0] [1 0 1] [0]
0() = [1]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [1 2 1] x3 + [0 0 0] x4 + [1]
[0 1 0] [0 0 0] [1 1 2] [0 0 0] [0]
[0 0 1] [0 0 0] [1 0 1] [0 0 1] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [3]
[3 3 3] [3 3 3] [3]
[3 3 3] [3 3 3] [3]
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]
natsFrom^#(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]
fst^#(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]
snd^#(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]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
head^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [1 1 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {10}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [3]
[0 1 2] [2]
[0 0 1] [3]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [1]
[3 1 1] [0 0 0] [2]
[2 0 0] [1 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [2]
[3 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [2]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [3 0 1] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
snd^#(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]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_9(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{2}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 1 1] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [2]
[0 0 1] [0 0 1] [1]
s(x1) = [1 1 0] x1 + [0]
[0 1 3] [0]
[0 0 1] [0]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 1 1] x1 + [0 1 1] x2 + [1]
[1 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
0() = [0]
[0]
[0]
nil() = [0]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 1] x1 + [0 0 0] x2 + [0 1 1] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
snd^#(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]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(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_9(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
afterNth^#(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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {11}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {1},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [3]
[0 1 2] [2]
[0 0 1] [3]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [0 0 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [1]
[3 1 1] [0 0 0] [2]
[2 0 0] [1 0 0] [1]
0() = [0]
[0]
[0]
nil() = [3]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [2]
[3 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1 0 0] [0 0 0] [2]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(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]
snd^#(x1) = [3 0 1] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {11}->{3}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {1},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 1 1] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [2]
[0 0 1] [0 0 1] [1]
s(x1) = [1 1 0] x1 + [0]
[0 1 3] [0]
[0 0 1] [0]
fst(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pair(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
snd(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt(x1, x2) = [0 1 1] x1 + [0 1 1] x2 + [1]
[1 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [1]
0() = [0]
[0]
[0]
nil() = [0]
[0]
[0]
u(x1, x2, x3, x4) = [1 0 1] x1 + [0 0 0] x2 + [0 1 1] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [1]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(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]
afterNth(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]
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]
natsFrom^#(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]
fst^#(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]
snd^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
splitAt^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
u^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_5(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(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_8(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
afterNth^#(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_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
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: natsFrom^#(N) -> c_0(N, natsFrom^#(s(N)))
, 2: fst^#(pair(XS, YS)) -> c_1(XS)
, 3: snd^#(pair(XS, YS)) -> c_2(YS)
, 4: splitAt^#(0(), XS) -> c_3(XS)
, 5: splitAt^#(s(N), cons(X, XS)) ->
c_4(u^#(splitAt(N, XS), N, X, XS))
, 6: u^#(pair(YS, ZS), N, X, XS) -> c_5(X, YS, ZS)
, 7: head^#(cons(N, XS)) -> c_6(N)
, 8: tail^#(cons(N, XS)) -> c_7(XS)
, 9: sel^#(N, XS) -> c_8(head^#(afterNth(N, XS)))
, 10: take^#(N, XS) -> c_9(fst^#(splitAt(N, XS)))
, 11: afterNth^#(N, XS) -> c_10(snd^#(splitAt(N, XS)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ NA ]
|
`->{3} [ NA ]
->{10} [ NA ]
|
`->{2} [ NA ]
->{9} [ NA ]
|
`->{7} [ NA ]
->{8} [ YES(?,O(n^2)) ]
->{5} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(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(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {2},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
u(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_0(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(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: {natsFrom^#(N) -> c_0(N, natsFrom^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
u(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [1 1] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {splitAt^#(0(), XS) -> c_3(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(splitAt^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
splitAt^#(x1, x2) = [2 2] x1 + [7 7] x2 + [7]
[2 2] [7 7] [3]
c_3(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {5}: NA
------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [1 3] x2 + [1]
[0 1] [0 0] [0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [1 0] x1 + [3 0] x2 + [2]
[0 0] [3 2] [2]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [2]
[0 0] [0 0] [1 3] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [1 3] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
u^#(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 1] x3 + [0 0] x4 + [0]
[3 3] [3 3] [3 3] [3 3] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {5}->{6}: NA
-----------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 2] x1 + [0 3] x2 + [1]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [3]
[0 0] [0 0] [0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
u^#(x1, x2, x3, x4) = [3 3] x1 + [3 3] x2 + [0 0] x3 + [3 3] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [1 1] x1 + [1 1] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(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]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
u(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(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: {tail^#(cons(N, XS)) -> c_7(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
tail^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_7(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {9}: NA
------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {1}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 3] x2 + [3]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 1] x2 + [1]
[0 0] [0 1] [0]
snd(x1) = [1 0] x1 + [0]
[0 1] [3]
splitAt(x1, x2) = [0 0] x1 + [1 1] x2 + [2]
[0 0] [3 1] [0]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [2]
[2 0] [0 0] [1 0] [0 2] [2]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [1 1] x2 + [3]
[0 2] [3 2] [3]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [2 0] x1 + [0]
[3 3] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {9}->{7}: NA
-----------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {1}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [1]
[0 0] [0 1] [2]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [2]
snd(x1) = [1 0] x1 + [0]
[0 1] [0]
splitAt(x1, x2) = [0 0] x1 + [1 3] x2 + [2]
[0 0] [0 1] [3]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 2] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [0]
[0 1] [0 0] [0 0] [0 0] [2]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [3 3] x1 + [3 3] x2 + [3]
[3 3] [3 3] [3]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {10}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [3]
[0 0] [0 1] [3]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 1] x1 + [3 3] x2 + [2]
[0 1] [0 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [2]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [1 1] x1 + [0]
[3 3] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{2}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 1] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {1},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [3]
[0 0] [0 1] [3]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 1] x1 + [3 3] x2 + [2]
[0 1] [0 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [2]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [1 1] x1 + [0]
[3 3] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {11}->{3}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {1},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
snd(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [3]
[0]
u(x1, x2, x3, x4) = [1 1] x1 + [0 0] x2 + [1 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
snd^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
u^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: natsFrom^#(N) -> c_0(N, natsFrom^#(s(N)))
, 2: fst^#(pair(XS, YS)) -> c_1(XS)
, 3: snd^#(pair(XS, YS)) -> c_2(YS)
, 4: splitAt^#(0(), XS) -> c_3(XS)
, 5: splitAt^#(s(N), cons(X, XS)) ->
c_4(u^#(splitAt(N, XS), N, X, XS))
, 6: u^#(pair(YS, ZS), N, X, XS) -> c_5(X, YS, ZS)
, 7: head^#(cons(N, XS)) -> c_6(N)
, 8: tail^#(cons(N, XS)) -> c_7(XS)
, 9: sel^#(N, XS) -> c_8(head^#(afterNth(N, XS)))
, 10: take^#(N, XS) -> c_9(fst^#(splitAt(N, XS)))
, 11: afterNth^#(N, XS) -> c_10(snd^#(splitAt(N, XS)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ NA ]
|
`->{3} [ NA ]
->{10} [ NA ]
|
`->{2} [ NA ]
->{9} [ NA ]
|
`->{7} [ NA ]
->{8} [ YES(?,O(n^1)) ]
->{5} [ NA ]
|
`->{6} [ NA ]
->{4} [ YES(?,O(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(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {2},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
u(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [3] x1 + [0]
c_0(x1, x2) = [2] x1 + [1] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(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: {natsFrom^#(N) -> c_0(N, natsFrom^#(s(N)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
u(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {splitAt^#(0(), XS) -> c_3(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(splitAt^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [5]
splitAt^#(x1, x2) = [3] x1 + [7] x2 + [0]
c_3(x1) = [1] x1 + [0]
* Path {5}: NA
------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [2] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [0] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
u^#(x1, x2, x3, x4) = [1] x1 + [3] x2 + [1] x3 + [1] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5}->{6}: NA
-----------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}, Uargs(u^#) = {1}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [1] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
u^#(x1, x2, x3, x4) = [3] x1 + [3] x2 + [0] x3 + [3] x4 + [0]
c_5(x1, x2, x3) = [1] x1 + [3] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
u(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [3] x1 + [0]
c_7(x1) = [1] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(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: {tail^#(cons(N, XS)) -> c_7(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_7(x1) = [1] x1 + [1]
* Path {9}: NA
------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {1}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [1]
snd(x1) = [1] x1 + [0]
splitAt(x1, x2) = [0] x1 + [3] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [3] x1 + [3] x2 + [3]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_8(x1) = [1] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {9}->{7}: NA
-----------------
The usable rules for this path are:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(XS, YS)) -> YS
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {1},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {1}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {1},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [1]
snd(x1) = [1] x1 + [0]
splitAt(x1, x2) = [0] x1 + [3] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [3] x1 + [3] x2 + [3]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [3] x1 + [0]
c_6(x1) = [1] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [1] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [1] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [3] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_9(x1) = [1] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{2}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {1}, Uargs(c_1) = {}, Uargs(snd^#) = {},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {1}, Uargs(afterNth^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [1] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [3] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [1] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: NA
-------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {1},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [1] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [3] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_10(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}->{3}: NA
------------------
The usable rules for this path are:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
, u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(natsFrom) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(fst) = {}, Uargs(pair) = {}, Uargs(snd) = {},
Uargs(splitAt) = {}, Uargs(u) = {1}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(sel) = {}, Uargs(afterNth) = {},
Uargs(take) = {}, Uargs(natsFrom^#) = {}, Uargs(c_0) = {},
Uargs(fst^#) = {}, Uargs(c_1) = {}, Uargs(snd^#) = {1},
Uargs(c_2) = {}, Uargs(splitAt^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(u^#) = {}, Uargs(c_5) = {},
Uargs(head^#) = {}, Uargs(c_6) = {}, Uargs(tail^#) = {},
Uargs(c_7) = {}, Uargs(sel^#) = {}, Uargs(c_8) = {},
Uargs(take^#) = {}, Uargs(c_9) = {}, Uargs(afterNth^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
natsFrom(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
s(x1) = [1] x1 + [0]
fst(x1) = [0] x1 + [0]
pair(x1, x2) = [0] x1 + [1] x2 + [0]
snd(x1) = [0] x1 + [0]
splitAt(x1, x2) = [0] x1 + [1] x2 + [2]
0() = [0]
nil() = [3]
u(x1, x2, x3, x4) = [1] x1 + [0] x2 + [1] x3 + [0] x4 + [2]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
sel(x1, x2) = [0] x1 + [0] x2 + [0]
afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
natsFrom^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
fst^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
snd^#(x1) = [3] x1 + [0]
c_2(x1) = [0] x1 + [0]
splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
u^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
head^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.