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:
{ pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
, zip(nil(), XS) -> nil()
, zip(X, nil()) -> nil()
, zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
, tail(cons(X, XS)) -> XS
, repItems(nil()) -> nil()
, repItems(cons(X, XS)) -> cons(X, cons(X, repItems(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: pairNs^#() -> c_0(incr^#(oddNs()))
, 2: oddNs^#() -> c_1(incr^#(pairNs()))
, 3: incr^#(cons(X, XS)) -> c_2(incr^#(XS))
, 4: take^#(0(), XS) -> c_3()
, 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
, 6: zip^#(nil(), XS) -> c_5()
, 7: zip^#(X, nil()) -> c_6()
, 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))
, 9: tail^#(cons(X, XS)) -> c_8()
, 10: repItems^#(nil()) -> c_9()
, 11: repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(n^3)) ]
|
|->{6} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^3)) ]
->{5} [ YES(?,O(n^3)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ inherited ]
|
`->{3} [ MAYBE ]
->{1} [ inherited ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}.
* Path {1}->{3}: NA
-----------------
The usable rules for this path are:
{ oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ oddNs^#() -> c_1(incr^#(pairNs()))
, incr^#(cons(X, XS)) -> c_2(incr^#(XS))
, pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
Proof Output:
The input cannot be shown compatible
* Path {5}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
[0]
cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
[0 1 0] [0 1 3] [0]
[0 0 0] [0 0 1] [0]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
take^#(x1, x2) = [0 4 0] x1 + [4 1 0] x2 + [0]
[0 0 0] [2 0 0] [0]
[4 0 0] [0 0 0] [0]
c_4(x1) = [1 0 0] x1 + [7]
[0 0 0] [0]
[0 0 0] [0]
* Path {5}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {take^#(0(), XS) -> c_3()}
Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 0 0] [0 0 0] [0]
0() = [2]
[0]
[0]
s(x1) = [1 4 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [2 0 0] x1 + [2 2 0] x2 + [0]
[2 0 0] [4 0 0] [0]
[0 0 0] [4 4 0] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
[0]
cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
[0 1 0] [0 1 3] [0]
[0 0 0] [0 0 1] [0]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [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: innermost DP runtime-complexity with respect to
Strict Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [2]
[0 0 0] [0 0 2] [2]
[0 0 0] [0 0 0] [0]
zip^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_7(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {8}->{6}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {zip^#(nil(), XS) -> c_5()}
Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 4 2] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 0 0] [0 0 1] [0]
nil() = [2]
[2]
[2]
zip^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_5() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {8}->{7}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8() = [0]
[0]
[0]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [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^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {zip^#(X, nil()) -> c_6()}
Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 0] x2 + [2]
[0 0 0] [0 1 3] [2]
[0 0 0] [0 0 1] [2]
nil() = [2]
[2]
[2]
zip^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_6() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {9}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [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(X, XS)) -> c_8()}
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_8() = [0]
[1]
[1]
* Path {11}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
[0]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
repItems^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 1 2] [2]
[0 0 0] [0 0 0] [0]
repItems^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_10(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {11}->{10}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {repItems^#(nil()) -> c_9()}
Weak Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 1] [1]
[0 0 0] [0 0 0] [0]
nil() = [2]
[2]
[2]
repItems^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_9() = [1]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: pairNs^#() -> c_0(incr^#(oddNs()))
, 2: oddNs^#() -> c_1(incr^#(pairNs()))
, 3: incr^#(cons(X, XS)) -> c_2(incr^#(XS))
, 4: take^#(0(), XS) -> c_3()
, 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
, 6: zip^#(nil(), XS) -> c_5()
, 7: zip^#(X, nil()) -> c_6()
, 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))
, 9: tail^#(cons(X, XS)) -> c_8()
, 10: repItems^#(nil()) -> c_9()
, 11: repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(n^2)) ]
|
|->{6} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{2} [ inherited ]
|
`->{3} [ NA ]
->{1} [ inherited ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}.
* Path {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ pairNs^#() -> c_0(incr^#(oddNs()))
, incr^#(cons(X, XS)) -> c_2(incr^#(XS))
, oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
Proof Output:
The input cannot be shown compatible
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: NA
-----------------
The usable rules for this path are:
{ pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
[0 0] [0 1] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
s(x1) = [1 4] x1 + [0]
[0 1] [2]
take^#(x1, x2) = [1 4] x1 + [0 1] x2 + [0]
[0 2] [0 0] [2]
c_4(x1) = [1 2] x1 + [3]
[0 0] [3]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {take^#(0(), XS) -> c_3()}
Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [2]
take^#(x1, x2) = [1 2] x1 + [0 2] x2 + [2]
[3 3] [0 4] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [1]
[2 0] [4]
* 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
[0 0] [0 1] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
zip^#(x1, x2) = [3 3] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 1] [2]
zip^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_7(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {8}->{6}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {zip^#(nil(), XS) -> c_5()}
Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 1] [0]
nil() = [2]
[2]
zip^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_5() = [1]
[0]
c_7(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {8}->{7}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {zip^#(X, nil()) -> c_6()}
Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 1] [0]
nil() = [2]
[0]
zip^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_6() = [1]
[0]
c_7(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {9}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [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(X, XS)) -> c_8()}
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_8() = [0]
[1]
* Path {11}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
repItems^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
repItems^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11}->{10}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {repItems^#(nil()) -> c_9()}
Weak Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [3]
nil() = [2]
[2]
repItems^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [5]
[2 0] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: pairNs^#() -> c_0(incr^#(oddNs()))
, 2: oddNs^#() -> c_1(incr^#(pairNs()))
, 3: incr^#(cons(X, XS)) -> c_2(incr^#(XS))
, 4: take^#(0(), XS) -> c_3()
, 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
, 6: zip^#(nil(), XS) -> c_5()
, 7: zip^#(X, nil()) -> c_6()
, 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))
, 9: tail^#(cons(X, XS)) -> c_8()
, 10: repItems^#(nil()) -> c_9()
, 11: repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{9} [ YES(?,O(1)) ]
->{8} [ YES(?,O(n^1)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ inherited ]
|
`->{3} [ NA ]
->{1} [ inherited ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}.
* Path {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ pairNs^#() -> c_0(incr^#(oddNs()))
, incr^#(cons(X, XS)) -> c_2(incr^#(XS))
, oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
Proof Output:
The input cannot be shown compatible
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: NA
-----------------
The usable rules for this path are:
{ pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
take^#(x1, x2) = [1] x1 + [3] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_8() = [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [2]
take^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_4(x1) = [1] x1 + [3]
* Path {5}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_8() = [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {take^#(0(), XS) -> c_3()}
Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [2]
0() = [2]
s(x1) = [1] x1 + [0]
take^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_3() = [1]
c_4(x1) = [1] x1 + [3]
* 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
zip^#(x1, x2) = [3] x1 + [1] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_8() = [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [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: innermost DP runtime-complexity with respect to
Strict Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [2]
zip^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7(x1) = [1] x1 + [7]
* Path {8}->{6}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_8() = [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {zip^#(nil(), XS) -> c_5()}
Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [2]
nil() = [2]
zip^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_5() = [1]
c_7(x1) = [1] x1 + [2]
* Path {8}->{7}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_8() = [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [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: innermost DP runtime-complexity with respect to
Strict Rules: {zip^#(X, nil()) -> c_6()}
Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [2]
nil() = [2]
zip^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_6() = [1]
c_7(x1) = [1] x1 + [7]
* Path {9}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_8() = [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [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(X, XS)) -> c_8()}
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_8() = [1]
* Path {11}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_8() = [0]
repItems^#(x1) = [3] x1 + [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [4]
repItems^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [7]
* Path {11}->{10}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_8() = [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {repItems^#(nil()) -> c_9()}
Weak Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [0]
nil() = [2]
repItems^#(x1) = [2] x1 + [0]
c_9() = [1]
c_10(x1) = [1] x1 + [0]
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:
{ pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
, zip(nil(), XS) -> nil()
, zip(X, nil()) -> nil()
, zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
, tail(cons(X, XS)) -> XS
, repItems(nil()) -> nil()
, repItems(cons(X, XS)) -> cons(X, cons(X, repItems(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: pairNs^#() -> c_0(incr^#(oddNs()))
, 2: oddNs^#() -> c_1(incr^#(pairNs()))
, 3: incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
, 4: take^#(0(), XS) -> c_3()
, 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
, 6: zip^#(nil(), XS) -> c_5()
, 7: zip^#(X, nil()) -> c_6()
, 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))
, 9: tail^#(cons(X, XS)) -> c_8(XS)
, 10: repItems^#(nil()) -> c_9()
, 11: repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^3)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{9} [ YES(?,O(n^3)) ]
->{8} [ YES(?,O(n^3)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{5} [ YES(?,O(n^3)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ inherited ]
|
`->{3} [ NA ]
->{1} [ inherited ]
|
`->{3} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}.
* Path {1}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ pairNs^#() -> c_0(incr^#(oddNs()))
, incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
, oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
Proof Output:
The input cannot be shown compatible
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: NA
-----------------
The usable rules for this path are:
{ pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
[0 1 3] [0 1 0] [0]
[0 0 1] [0 0 0] [0]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, 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]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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]
tail^#(x1) = [0 0 0] x1 + [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]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [1 2 2] x2 + [0]
[0 0 2] [0 1 0] [0]
[0 0 0] [0 0 1] [2]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 0] [0]
take^#(x1, x2) = [2 2 0] x1 + [3 3 2] x2 + [0]
[2 3 0] [2 5 4] [0]
[4 0 0] [3 2 2] [0]
c_4(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [7]
[0 0 0] [0 0 0] [3]
[1 0 0] [0 0 0] [0]
* Path {5}->{4}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, 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]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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]
tail^#(x1) = [0 0 0] x1 + [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]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(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]
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: {take^#(0(), XS) -> c_3()}
Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [1 6 0] x2 + [2]
[0 0 2] [0 0 0] [2]
[0 0 0] [0 0 0] [0]
0() = [2]
[2]
[2]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
take^#(x1, x2) = [2 2 2] x1 + [2 2 0] x2 + [0]
[2 3 1] [2 2 0] [0]
[2 2 0] [2 2 0] [0]
c_3() = [1]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [7]
[0 0 0] [0 0 0] [4]
[0 0 0] [0 0 0] [7]
* 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [1 0 0] x2 + [0]
[0 1 3] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, 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]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0]
tail^#(x1) = [0 0 0] x1 + [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]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(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]
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:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [1 4 0] x2 + [4]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
zip^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [0]
[2 0 0] [0 0 0] [0]
[2 0 0] [0 0 0] [0]
c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [7]
[0 0 0] [0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [0 0 0] [6]
* Path {8}->{6}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, 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]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0]
tail^#(x1) = [0 0 0] x1 + [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]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(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]
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: {zip^#(nil(), XS) -> c_5()}
Weak Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [1 0 0] x2 + [2]
[0 0 2] [0 1 0] [2]
[0 0 0] [0 0 1] [2]
nil() = [2]
[2]
[2]
zip^#(x1, x2) = [2 2 2] x1 + [0 1 0] x2 + [0]
[2 2 0] [0 0 0] [0]
[2 0 2] [0 2 0] [0]
c_5() = [1]
[0]
[0]
c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [1]
[0 0 0] [0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [0 0 0] [7]
* Path {8}->{7}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, 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]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0]
tail^#(x1) = [0 0 0] x1 + [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]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zip^#(X, nil()) -> c_6()}
Weak Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 0 2] x1 + [1 2 0] x2 + [0]
[0 0 2] [0 0 2] [0]
[0 0 1] [0 0 1] [0]
nil() = [2]
[2]
[2]
zip^#(x1, x2) = [2 4 0] x1 + [1 2 2] x2 + [0]
[2 2 0] [2 3 2] [0]
[2 2 0] [2 3 2] [0]
c_6() = [1]
[0]
[0]
c_7(x1, x2, x3) = [0 0 0] x1 + [1 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
* Path {9}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, 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]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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]
tail^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(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]
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(X, XS)) -> c_8(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {}
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_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {11}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
[0 1 3] [0 1 0] [0]
[0 0 1] [0 0 0] [0]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, 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]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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]
tail^#(x1) = [0 0 0] x1 + [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]
repItems^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_9() = [0]
[0]
[0]
c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [1 4 2] x2 + [2]
[0 0 2] [0 1 1] [2]
[0 0 0] [0 0 1] [2]
repItems^#(x1) = [2 1 3] x1 + [0]
[0 2 2] [0]
[2 2 0] [0]
c_10(x1, x2, x3) = [0 4 3] x1 + [2 0 3] x2 + [1 0 0] x3 + [7]
[0 0 4] [0 0 0] [0 0 0] [6]
[2 4 3] [0 0 4] [0 0 0] [0]
* Path {11}->{10}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [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]
0() = [0]
[0]
[0]
incr(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs() = [0]
[0]
[0]
s(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]
nil() = [0]
[0]
[0]
zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
repItems(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pairNs^#() = [0]
[0]
[0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
incr^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
oddNs^#() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, 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]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
zip^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(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]
tail^#(x1) = [0 0 0] x1 + [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]
repItems^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 1 0] [0]
[0 0 0] [0 0 0] [0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {repItems^#(nil()) -> c_9()}
Weak Rules: {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
[0 0 0] [0 0 4] [0]
[0 0 0] [0 0 0] [0]
nil() = [2]
[2]
[2]
repItems^#(x1) = [1 2 2] x1 + [0]
[3 2 2] [0]
[3 2 2] [0]
c_9() = [1]
[0]
[0]
c_10(x1, x2, x3) = [0 0 0] x1 + [1 0 0] x2 + [1 0 0] x3 + [0]
[3 0 0] [0 0 0] [2 0 0] [0]
[1 0 0] [2 0 0] [0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: pairNs^#() -> c_0(incr^#(oddNs()))
, 2: oddNs^#() -> c_1(incr^#(pairNs()))
, 3: incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
, 4: take^#(0(), XS) -> c_3()
, 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
, 6: zip^#(nil(), XS) -> c_5()
, 7: zip^#(X, nil()) -> c_6()
, 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))
, 9: tail^#(cons(X, XS)) -> c_8(XS)
, 10: repItems^#(nil()) -> c_9()
, 11: repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{9} [ YES(?,O(n^2)) ]
->{8} [ YES(?,O(n^2)) ]
|
|->{6} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{2} [ inherited ]
|
`->{3} [ MAYBE ]
->{1} [ inherited ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}.
* Path {1}->{3}: NA
-----------------
The usable rules for this path are:
{ oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ oddNs^#() -> c_1(incr^#(pairNs()))
, incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
, pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
take^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [3]
[0 0] [0 1] [2]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
take^#(x1, x2) = [0 0] x1 + [2 3] x2 + [0]
[0 1] [3 3] [0]
c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [5]
[1 0] [0 0] [7]
* Path {5}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(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]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {take^#(0(), XS) -> c_3()}
Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 0] [0 1] [0]
0() = [3]
[2]
s(x1) = [1 2] x1 + [3]
[0 1] [0]
take^#(x1, x2) = [2 1] x1 + [2 1] x2 + [0]
[3 2] [4 1] [0]
c_3() = [1]
[1]
c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [6]
* 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(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]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
zip^#(x1, x2) = [2 1] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1, x2, x3) = [0 1] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 1] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 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:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
[0 0] [0 0] [2]
zip^#(x1, x2) = [2 4] x1 + [0 0] x2 + [0]
[0 4] [2 0] [0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [5]
[0 0] [0 0] [0 0] [6]
* Path {8}->{6}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(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]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 1] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zip^#(nil(), XS) -> c_5()}
Weak Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
[0 0] [0 1] [2]
nil() = [2]
[2]
zip^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[2 2] [2 1] [0]
c_5() = [1]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [6]
[0 0] [0 0] [0 0] [7]
* Path {8}->{7}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(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]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 1] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {zip^#(X, nil()) -> c_6()}
Weak Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
[0 0] [0 1] [2]
nil() = [2]
[0]
zip^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[2 2] [2 1] [0]
c_6() = [1]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [7]
[0 0] [0 0] [0 0] [7]
* Path {9}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(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]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tail^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 0] [0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 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(X, XS)) -> c_8(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {}
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_8(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {11}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(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]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_9() = [0]
[0]
c_10(x1, x2, x3) = [1 3] x1 + [0 3] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 2] x2 + [1]
[0 0] [0 1] [2]
repItems^#(x1) = [2 4] x1 + [0]
[2 3] [0]
c_10(x1, x2, x3) = [0 4] x1 + [2 0] x2 + [1 0] x3 + [1]
[0 4] [2 0] [0 0] [0]
* Path {11}->{10}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
incr(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems(x1) = [0 0] x1 + [0]
[0 0] [0]
pairNs^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
incr^#(x1) = [0 0] x1 + [0]
[0 0] [0]
oddNs^#() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(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]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
repItems^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 1] [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: {repItems^#(nil()) -> c_9()}
Weak Rules: {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 3] x1 + [1 1] x2 + [2]
[0 0] [0 0] [0]
nil() = [0]
[2]
repItems^#(x1) = [5 2] x1 + [0]
[4 2] [0]
c_9() = [1]
[0]
c_10(x1, x2, x3) = [0 7] x1 + [3 7] x2 + [1 0] x3 + [7]
[2 7] [1 5] [0 0] [6]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: pairNs^#() -> c_0(incr^#(oddNs()))
, 2: oddNs^#() -> c_1(incr^#(pairNs()))
, 3: incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
, 4: take^#(0(), XS) -> c_3()
, 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
, 6: zip^#(nil(), XS) -> c_5()
, 7: zip^#(X, nil()) -> c_6()
, 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))
, 9: tail^#(cons(X, XS)) -> c_8(XS)
, 10: repItems^#(nil()) -> c_9()
, 11: repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{9} [ YES(?,O(n^1)) ]
->{8} [ YES(?,O(n^1)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{5} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ inherited ]
|
`->{3} [ MAYBE ]
->{1} [ inherited ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}.
* Path {1}->{3}: NA
-----------------
The usable rules for this path are:
{ oddNs() -> incr(pairNs())
, pairNs() -> cons(0(), incr(oddNs()))
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3}.
* Path {2}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ oddNs^#() -> c_1(incr^#(pairNs()))
, incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
, pairNs() -> cons(0(), incr(oddNs()))
, oddNs() -> incr(pairNs())
, incr(cons(X, XS)) -> cons(s(X), incr(XS))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [1] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4(x1, x2) = [1] x1 + [1] x2 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tail^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [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: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [2]
s(x1) = [1] x1 + [2]
take^#(x1, x2) = [2] x1 + [3] x2 + [0]
c_4(x1, x2) = [1] x1 + [1] x2 + [1]
* Path {5}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [1] x2 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tail^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [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: {take^#(0(), XS) -> c_3()}
Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
Uargs(c_4) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [2]
0() = [2]
s(x1) = [1] x1 + [2]
take^#(x1, x2) = [3] x1 + [2] x2 + [2]
c_3() = [1]
c_4(x1, x2) = [0] x1 + [1] x2 + [5]
* 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
zip^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
tail^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [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:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [2]
zip^#(x1, x2) = [4] x1 + [2] x2 + [0]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [7]
* Path {8}->{6}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
tail^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [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: {zip^#(nil(), XS) -> c_5()}
Weak Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [2]
zip^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_5() = [1]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [7]
* Path {8}->{7}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
tail^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [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: {zip^#(X, nil()) -> c_6()}
Weak Rules:
{zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [2]
nil() = [2]
zip^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_6() = [1]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [7]
* Path {9}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tail^#(x1) = [3] x1 + [0]
c_8(x1) = [1] x1 + [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [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(X, XS)) -> c_8(XS)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_8(x1) = [1] x1 + [1]
* Path {11}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tail^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
repItems^#(x1) = [3] x1 + [0]
c_9() = [0]
c_10(x1, x2, x3) = [2] x1 + [1] x2 + [1] x3 + [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:
{repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [4]
repItems^#(x1) = [2] x1 + [0]
c_10(x1, x2, x3) = [2] x1 + [0] x2 + [1] x3 + [7]
* Path {11}->{10}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pairNs() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
incr(x1) = [0] x1 + [0]
oddNs() = [0]
s(x1) = [0] x1 + [0]
take(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
zip(x1, x2) = [0] x1 + [0] x2 + [0]
pair(x1, x2) = [0] x1 + [0] x2 + [0]
tail(x1) = [0] x1 + [0]
repItems(x1) = [0] x1 + [0]
pairNs^#() = [0]
c_0(x1) = [0] x1 + [0]
incr^#(x1) = [0] x1 + [0]
oddNs^#() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
take^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tail^#(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
repItems^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [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: {repItems^#(nil()) -> c_9()}
Weak Rules: {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [0]
nil() = [0]
repItems^#(x1) = [0] x1 + [1]
c_9() = [0]
c_10(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
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.