Tool CaT
stdout:
MAYBE
Problem:
plus(x,y) -> plusIter(x,y,0())
plusIter(x,y,z) -> ifPlus(le(x,z),x,y,z)
ifPlus(true(),x,y,z) -> y
ifPlus(false(),x,y,z) -> plusIter(x,s(y),s(z))
le(s(x),0()) -> false()
le(0(),y) -> true()
le(s(x),s(y)) -> le(x,y)
sum(xs) -> sumIter(xs,0())
sumIter(xs,x) -> ifSum(isempty(xs),xs,x,plus(x,head(xs)))
ifSum(true(),xs,x,y) -> x
ifSum(false(),xs,x,y) -> sumIter(tail(xs),y)
isempty(nil()) -> true()
isempty(cons(x,xs)) -> false()
head(nil()) -> error()
head(cons(x,xs)) -> x
tail(nil()) -> nil()
tail(cons(x,xs)) -> xs
a() -> b()
a() -> c()
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus(x, y) -> plusIter(x, y, 0())
, plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
, ifPlus(true(), x, y, z) -> y
, ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, sum(xs) -> sumIter(xs, 0())
, sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
, ifSum(true(), xs, x, y) -> x
, ifSum(false(), xs, x, y) -> sumIter(tail(xs), y)
, isempty(nil()) -> true()
, isempty(cons(x, xs)) -> false()
, head(nil()) -> error()
, head(cons(x, xs)) -> x
, tail(nil()) -> nil()
, tail(cons(x, xs)) -> xs
, a() -> b()
, a() -> c()}
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: plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, 2: plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, 3: ifPlus^#(true(), x, y, z) -> c_2()
, 4: ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, 5: le^#(s(x), 0()) -> c_4()
, 6: le^#(0(), y) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: sum^#(xs) -> c_7(sumIter^#(xs, 0()))
, 9: sumIter^#(xs, x) ->
c_8(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
, 10: ifSum^#(true(), xs, x, y) -> c_9()
, 11: ifSum^#(false(), xs, x, y) -> c_10(sumIter^#(tail(xs), y))
, 12: isempty^#(nil()) -> c_11()
, 13: isempty^#(cons(x, xs)) -> c_12()
, 14: head^#(nil()) -> c_13()
, 15: head^#(cons(x, xs)) -> c_14()
, 16: tail^#(nil()) -> c_15()
, 17: tail^#(cons(x, xs)) -> c_16()
, 18: a^#() -> c_17()
, 19: a^#() -> c_18()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ YES(?,O(1)) ]
->{18} [ YES(?,O(1)) ]
->{17} [ YES(?,O(1)) ]
->{16} [ YES(?,O(1)) ]
->{15} [ YES(?,O(1)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(1)) ]
->{12} [ YES(?,O(1)) ]
->{8} [ inherited ]
|
`->{9,11} [ inherited ]
|
`->{10} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^3)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2,4} [ MAYBE ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,4}.
* Path {1}->{2,4}: MAYBE
----------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2,4}->{3}: NA
------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {1}, Uargs(plusIter^#) = {},
Uargs(c_1) = {1}, Uargs(ifPlus^#) = {1}, Uargs(c_3) = {1},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [1]
[0]
[0]
false() = [1]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
plusIter^#(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]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
ifPlus^#(x1, x2, x3, x4) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
le^#(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_6(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {le^#(s(x), 0()) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
le^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_4() = [1]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {le^#(0(), y) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
le^#(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_6(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}->{10}: NA
--------------------------
The usable rules for this path are:
{ plus(x, y) -> plusIter(x, y, 0())
, isempty(nil()) -> true()
, isempty(cons(x, xs)) -> false()
, head(nil()) -> error()
, head(cons(x, xs)) -> x
, tail(nil()) -> nil()
, tail(cons(x, xs)) -> xs
, plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
, ifPlus(true(), x, y, z) -> y
, ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {12}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {isempty^#(nil()) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isempty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
isempty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_11() = [0]
[1]
[1]
* Path {13}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {isempty^#(cons(x, xs)) -> c_12()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(isempty^#) = {}
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]
isempty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_12() = [0]
[1]
[1]
* Path {14}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {head^#(nil()) -> c_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
head^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_13() = [0]
[1]
[1]
* Path {15}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {head^#(cons(x, xs)) -> c_14()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}
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]
head^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_14() = [0]
[1]
[1]
* Path {16}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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^#(nil()) -> c_15()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
tail^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_15() = [0]
[1]
[1]
* Path {17}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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_16()}
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_16() = [0]
[1]
[1]
* Path {18}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {a^#() -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
[7]
c_17() = [0]
[3]
[3]
* Path {19}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16() = [0]
[0]
[0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {a^#() -> c_18()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
[7]
c_18() = [0]
[3]
[3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, 2: plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, 3: ifPlus^#(true(), x, y, z) -> c_2()
, 4: ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, 5: le^#(s(x), 0()) -> c_4()
, 6: le^#(0(), y) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: sum^#(xs) -> c_7(sumIter^#(xs, 0()))
, 9: sumIter^#(xs, x) ->
c_8(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
, 10: ifSum^#(true(), xs, x, y) -> c_9()
, 11: ifSum^#(false(), xs, x, y) -> c_10(sumIter^#(tail(xs), y))
, 12: isempty^#(nil()) -> c_11()
, 13: isempty^#(cons(x, xs)) -> c_12()
, 14: head^#(nil()) -> c_13()
, 15: head^#(cons(x, xs)) -> c_14()
, 16: tail^#(nil()) -> c_15()
, 17: tail^#(cons(x, xs)) -> c_16()
, 18: a^#() -> c_17()
, 19: a^#() -> c_18()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ YES(?,O(1)) ]
->{18} [ YES(?,O(1)) ]
->{17} [ YES(?,O(1)) ]
->{16} [ YES(?,O(1)) ]
->{15} [ YES(?,O(1)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(1)) ]
->{12} [ YES(?,O(1)) ]
->{8} [ inherited ]
|
`->{9,11} [ inherited ]
|
`->{10} [ NA ]
->{7} [ YES(?,O(n^2)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2,4} [ MAYBE ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,4}.
* Path {1}->{2,4}: MAYBE
----------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2,4}->{3}: NA
------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {1}, Uargs(plusIter^#) = {},
Uargs(c_1) = {1}, Uargs(ifPlus^#) = {1}, Uargs(c_3) = {1},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [1]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [2 0] x1 + [3 0] x2 + [3]
[0 0] [0 0] [3]
true() = [0]
[1]
false() = [1]
[1]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
ifPlus^#(x1, x2, x3, x4) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_6(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), 0()) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_4() = [1]
[0]
c_6(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {le^#(0(), y) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_5() = [1]
[0]
c_6(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}->{10}: NA
--------------------------
The usable rules for this path are:
{ plus(x, y) -> plusIter(x, y, 0())
, isempty(nil()) -> true()
, isempty(cons(x, xs)) -> false()
, head(nil()) -> error()
, head(cons(x, xs)) -> x
, tail(nil()) -> nil()
, tail(cons(x, xs)) -> xs
, plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
, ifPlus(true(), x, y, z) -> y
, ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {12}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {isempty^#(nil()) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isempty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
isempty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_11() = [0]
[1]
* Path {13}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {isempty^#(cons(x, xs)) -> c_12()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(isempty^#) = {}
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]
isempty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_12() = [0]
[1]
* Path {14}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {head^#(nil()) -> c_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
head^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_13() = [0]
[1]
* Path {15}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {head^#(cons(x, xs)) -> c_14()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}
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]
head^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_14() = [0]
[1]
* Path {16}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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^#(nil()) -> c_15()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
tail^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_15() = [0]
[1]
* Path {17}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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_16()}
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_16() = [0]
[1]
* Path {18}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {a^#() -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
c_17() = [0]
[1]
* Path {19}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16() = [0]
[0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {a^#() -> c_18()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
c_18() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, 2: plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, 3: ifPlus^#(true(), x, y, z) -> c_2()
, 4: ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, 5: le^#(s(x), 0()) -> c_4()
, 6: le^#(0(), y) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: sum^#(xs) -> c_7(sumIter^#(xs, 0()))
, 9: sumIter^#(xs, x) ->
c_8(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
, 10: ifSum^#(true(), xs, x, y) -> c_9()
, 11: ifSum^#(false(), xs, x, y) -> c_10(sumIter^#(tail(xs), y))
, 12: isempty^#(nil()) -> c_11()
, 13: isempty^#(cons(x, xs)) -> c_12()
, 14: head^#(nil()) -> c_13()
, 15: head^#(cons(x, xs)) -> c_14()
, 16: tail^#(nil()) -> c_15()
, 17: tail^#(cons(x, xs)) -> c_16()
, 18: a^#() -> c_17()
, 19: a^#() -> c_18()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ YES(?,O(1)) ]
->{18} [ YES(?,O(1)) ]
->{17} [ YES(?,O(1)) ]
->{16} [ YES(?,O(1)) ]
->{15} [ YES(?,O(1)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(1)) ]
->{12} [ YES(?,O(1)) ]
->{8} [ inherited ]
|
`->{9,11} [ inherited ]
|
`->{10} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
`->{2,4} [ MAYBE ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,4}.
* Path {1}->{2,4}: MAYBE
----------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2,4}->{3}: NA
------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {1}, Uargs(plusIter^#) = {},
Uargs(c_1) = {1}, Uargs(ifPlus^#) = {1}, Uargs(c_3) = {1},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [2] x2 + [1]
true() = [0]
false() = [0]
s(x1) = [1] x1 + [2]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [1] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {le^#(s(x), 0()) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [1]
c_6(x1) = [1] x1 + [7]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {le^#(0(), y) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_5() = [1]
c_6(x1) = [1] x1 + [2]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}->{10}: NA
--------------------------
The usable rules for this path are:
{ plus(x, y) -> plusIter(x, y, 0())
, isempty(nil()) -> true()
, isempty(cons(x, xs)) -> false()
, head(nil()) -> error()
, head(cons(x, xs)) -> x
, tail(nil()) -> nil()
, tail(cons(x, xs)) -> xs
, plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
, ifPlus(true(), x, y, z) -> y
, ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {12}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {isempty^#(nil()) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isempty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
isempty^#(x1) = [1] x1 + [7]
c_11() = [1]
* Path {13}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {isempty^#(cons(x, xs)) -> c_12()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(isempty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
isempty^#(x1) = [1] x1 + [7]
c_12() = [1]
* Path {14}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {head^#(nil()) -> c_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
head^#(x1) = [1] x1 + [7]
c_13() = [1]
* Path {15}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {head^#(cons(x, xs)) -> c_14()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_14() = [1]
* Path {16}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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^#(nil()) -> c_15()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
tail^#(x1) = [1] x1 + [7]
c_15() = [1]
* Path {17}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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_16()}
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_16() = [1]
* Path {18}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {a^#() -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
c_17() = [0]
* Path {19}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(sum^#) = {},
Uargs(c_7) = {}, Uargs(sumIter^#) = {}, Uargs(c_8) = {},
Uargs(ifSum^#) = {}, Uargs(c_10) = {}, Uargs(isempty^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16() = [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {a^#() -> c_18()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
c_18() = [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:
{ plus(x, y) -> plusIter(x, y, 0())
, plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
, ifPlus(true(), x, y, z) -> y
, ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)
, sum(xs) -> sumIter(xs, 0())
, sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
, ifSum(true(), xs, x, y) -> x
, ifSum(false(), xs, x, y) -> sumIter(tail(xs), y)
, isempty(nil()) -> true()
, isempty(cons(x, xs)) -> false()
, head(nil()) -> error()
, head(cons(x, xs)) -> x
, tail(nil()) -> nil()
, tail(cons(x, xs)) -> xs
, a() -> b()
, a() -> c()}
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: plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, 2: plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, 3: ifPlus^#(true(), x, y, z) -> c_2(y)
, 4: ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, 5: le^#(s(x), 0()) -> c_4()
, 6: le^#(0(), y) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: sum^#(xs) -> c_7(sumIter^#(xs, 0()))
, 9: sumIter^#(xs, x) ->
c_8(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
, 10: ifSum^#(true(), xs, x, y) -> c_9(x)
, 11: ifSum^#(false(), xs, x, y) -> c_10(sumIter^#(tail(xs), y))
, 12: isempty^#(nil()) -> c_11()
, 13: isempty^#(cons(x, xs)) -> c_12()
, 14: head^#(nil()) -> c_13()
, 15: head^#(cons(x, xs)) -> c_14(x)
, 16: tail^#(nil()) -> c_15()
, 17: tail^#(cons(x, xs)) -> c_16(xs)
, 18: a^#() -> c_17()
, 19: a^#() -> c_18()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ YES(?,O(1)) ]
->{18} [ YES(?,O(1)) ]
->{17} [ YES(?,O(n^3)) ]
->{16} [ YES(?,O(1)) ]
->{15} [ YES(?,O(n^3)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(1)) ]
->{12} [ YES(?,O(1)) ]
->{8} [ inherited ]
|
`->{9,11} [ inherited ]
|
`->{10} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^3)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2,4} [ MAYBE ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,4}.
* Path {1}->{2,4}: MAYBE
----------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2,4}->{3}: NA
------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {1}, Uargs(plusIter^#) = {},
Uargs(c_1) = {1}, Uargs(ifPlus^#) = {1}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [1]
[0]
[0]
false() = [1]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
plusIter^#(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]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
ifPlus^#(x1, x2, x3, x4) = [3 0 0] x1 + [3 3 3] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
le^#(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_6(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {le^#(s(x), 0()) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
le^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_4() = [1]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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: {le^#(0(), y) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
le^#(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_6(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}->{10}: NA
--------------------------
The usable rules for this path are:
{ plus(x, y) -> plusIter(x, y, 0())
, isempty(nil()) -> true()
, isempty(cons(x, xs)) -> false()
, head(nil()) -> error()
, head(cons(x, xs)) -> x
, tail(nil()) -> nil()
, tail(cons(x, xs)) -> xs
, plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
, ifPlus(true(), x, y, z) -> y
, ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {12}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isempty^#(nil()) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isempty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
isempty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_11() = [0]
[1]
[1]
* Path {13}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isempty^#(cons(x, xs)) -> c_12()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(isempty^#) = {}
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]
isempty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_12() = [0]
[1]
[1]
* Path {14}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(nil()) -> c_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
head^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_13() = [0]
[1]
[1]
* Path {15}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(x1) = [1 0 1] 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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(cons(x, xs)) -> c_14(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [0 0 0] x2 + [2]
[0 0 2] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
head^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {16}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(nil()) -> c_15()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
tail^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_15() = [0]
[1]
[1]
* Path {17}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_15() = [0]
[0]
[0]
c_16(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [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_16(xs)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
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_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {18}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {a^#() -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
[7]
c_17() = [0]
[3]
[3]
* Path {19}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
plusIter(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]
0() = [0]
[0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
isempty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [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]
error() = [0]
[0]
[0]
a() = [0]
[0]
[0]
b() = [0]
[0]
[0]
c() = [0]
[0]
[0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plusIter^#(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]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sumIter^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isempty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(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_15() = [0]
[0]
[0]
c_16(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a^#() = [0]
[0]
[0]
c_17() = [0]
[0]
[0]
c_18() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {a^#() -> c_18()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
[7]
c_18() = [0]
[3]
[3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, 2: plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, 3: ifPlus^#(true(), x, y, z) -> c_2(y)
, 4: ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, 5: le^#(s(x), 0()) -> c_4()
, 6: le^#(0(), y) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: sum^#(xs) -> c_7(sumIter^#(xs, 0()))
, 9: sumIter^#(xs, x) ->
c_8(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
, 10: ifSum^#(true(), xs, x, y) -> c_9(x)
, 11: ifSum^#(false(), xs, x, y) -> c_10(sumIter^#(tail(xs), y))
, 12: isempty^#(nil()) -> c_11()
, 13: isempty^#(cons(x, xs)) -> c_12()
, 14: head^#(nil()) -> c_13()
, 15: head^#(cons(x, xs)) -> c_14(x)
, 16: tail^#(nil()) -> c_15()
, 17: tail^#(cons(x, xs)) -> c_16(xs)
, 18: a^#() -> c_17()
, 19: a^#() -> c_18()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ YES(?,O(1)) ]
->{18} [ YES(?,O(1)) ]
->{17} [ YES(?,O(n^2)) ]
->{16} [ YES(?,O(1)) ]
->{15} [ YES(?,O(n^2)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(1)) ]
->{12} [ YES(?,O(1)) ]
->{8} [ inherited ]
|
`->{9,11} [ inherited ]
|
`->{10} [ NA ]
->{7} [ YES(?,O(n^2)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2,4} [ MAYBE ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,4}.
* Path {1}->{2,4}: MAYBE
----------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2,4}->{3}: NA
------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {1}, Uargs(plusIter^#) = {},
Uargs(c_1) = {1}, Uargs(ifPlus^#) = {1}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [1 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
true() = [1]
[1]
false() = [1]
[1]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
ifPlus^#(x1, x2, x3, x4) = [3 0] x1 + [3 3] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [1 1] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_6(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {le^#(s(x), 0()) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_4() = [1]
[0]
c_6(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {le^#(0(), y) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_5() = [1]
[0]
c_6(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}->{10}: NA
--------------------------
The usable rules for this path are:
{ plus(x, y) -> plusIter(x, y, 0())
, isempty(nil()) -> true()
, isempty(cons(x, xs)) -> false()
, head(nil()) -> error()
, head(cons(x, xs)) -> x
, tail(nil()) -> nil()
, tail(cons(x, xs)) -> xs
, plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
, ifPlus(true(), x, y, z) -> y
, ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {12}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isempty^#(nil()) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isempty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
isempty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_11() = [0]
[1]
* Path {13}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isempty^#(cons(x, xs)) -> c_12()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(isempty^#) = {}
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]
isempty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_12() = [0]
[1]
* Path {14}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(nil()) -> c_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
head^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_13() = [0]
[1]
* Path {15}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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: {head^#(cons(x, xs)) -> c_14(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
head^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_14(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {16}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(nil()) -> c_15()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
tail^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_15() = [0]
[1]
* Path {17}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [1 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [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_16(xs)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
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_16(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {18}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {a^#() -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
c_17() = [0]
[1]
* Path {19}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plusIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
ifPlus(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ifSum(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
isempty(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
error() = [0]
[0]
a() = [0]
[0]
b() = [0]
[0]
c() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
plusIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
ifPlus^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
sumIter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
ifSum^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
isempty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
c_12() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15() = [0]
[0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
a^#() = [0]
[0]
c_17() = [0]
[0]
c_18() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {a^#() -> c_18()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
[7]
c_18() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, 2: plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, 3: ifPlus^#(true(), x, y, z) -> c_2(y)
, 4: ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, 5: le^#(s(x), 0()) -> c_4()
, 6: le^#(0(), y) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: sum^#(xs) -> c_7(sumIter^#(xs, 0()))
, 9: sumIter^#(xs, x) ->
c_8(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
, 10: ifSum^#(true(), xs, x, y) -> c_9(x)
, 11: ifSum^#(false(), xs, x, y) -> c_10(sumIter^#(tail(xs), y))
, 12: isempty^#(nil()) -> c_11()
, 13: isempty^#(cons(x, xs)) -> c_12()
, 14: head^#(nil()) -> c_13()
, 15: head^#(cons(x, xs)) -> c_14(x)
, 16: tail^#(nil()) -> c_15()
, 17: tail^#(cons(x, xs)) -> c_16(xs)
, 18: a^#() -> c_17()
, 19: a^#() -> c_18()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ YES(?,O(1)) ]
->{18} [ YES(?,O(1)) ]
->{17} [ YES(?,O(n^1)) ]
->{16} [ YES(?,O(1)) ]
->{15} [ YES(?,O(n^1)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(1)) ]
->{12} [ YES(?,O(1)) ]
->{8} [ inherited ]
|
`->{9,11} [ inherited ]
|
`->{10} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
`->{2,4} [ MAYBE ]
|
`->{3} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,4}.
* Path {1}->{2,4}: MAYBE
----------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ plus^#(x, y) -> c_0(plusIter^#(x, y, 0()))
, plusIter^#(x, y, z) -> c_1(ifPlus^#(le(x, z), x, y, z))
, ifPlus^#(false(), x, y, z) -> c_3(plusIter^#(x, s(y), s(z)))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{2,4}->{3}: NA
------------------------
The usable rules for this path are:
{ le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {1}, Uargs(plusIter^#) = {},
Uargs(c_1) = {1}, Uargs(ifPlus^#) = {1}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [2] x2 + [1]
true() = [0]
false() = [0]
s(x1) = [1] x1 + [2]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [1] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [3] x1 + [3] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [1] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {le^#(s(x), 0()) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [1]
c_6(x1) = [1] x1 + [7]
* Path {7}->{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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {le^#(0(), y) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_5() = [1]
c_6(x1) = [1] x1 + [2]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}: inherited
---------------------------
This path is subsumed by the proof of path {8}->{9,11}->{10}.
* Path {8}->{9,11}->{10}: NA
--------------------------
The usable rules for this path are:
{ plus(x, y) -> plusIter(x, y, 0())
, isempty(nil()) -> true()
, isempty(cons(x, xs)) -> false()
, head(nil()) -> error()
, head(cons(x, xs)) -> x
, tail(nil()) -> nil()
, tail(cons(x, xs)) -> xs
, plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
, ifPlus(true(), x, y, z) -> y
, ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
, le(s(x), 0()) -> false()
, le(0(), y) -> true()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {12}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isempty^#(nil()) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isempty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
isempty^#(x1) = [1] x1 + [7]
c_11() = [1]
* Path {13}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isempty^#(cons(x, xs)) -> c_12()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(isempty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
isempty^#(x1) = [1] x1 + [7]
c_12() = [1]
* Path {14}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(nil()) -> c_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
head^#(x1) = [1] x1 + [7]
c_13() = [1]
* Path {15}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [3] x1 + [0]
c_13() = [0]
c_14(x1) = [1] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [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: {head^#(cons(x, xs)) -> c_14(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_14(x1) = [1] x1 + [1]
* Path {16}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(nil()) -> c_15()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
tail^#(x1) = [1] x1 + [7]
c_15() = [1]
* Path {17}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [3] x1 + [0]
c_15() = [0]
c_16(x1) = [1] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [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_16(xs)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_16(x1) = [1] x1 + [1]
* Path {18}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {a^#() -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
c_17() = [0]
* Path {19}: 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(plus) = {}, Uargs(plusIter) = {}, Uargs(ifPlus) = {},
Uargs(le) = {}, Uargs(s) = {}, Uargs(sum) = {},
Uargs(sumIter) = {}, Uargs(ifSum) = {}, Uargs(isempty) = {},
Uargs(head) = {}, Uargs(tail) = {}, Uargs(cons) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(plusIter^#) = {},
Uargs(c_1) = {}, Uargs(ifPlus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(le^#) = {}, Uargs(c_6) = {},
Uargs(sum^#) = {}, Uargs(c_7) = {}, Uargs(sumIter^#) = {},
Uargs(c_8) = {}, Uargs(ifSum^#) = {}, Uargs(c_9) = {},
Uargs(c_10) = {}, Uargs(isempty^#) = {}, Uargs(head^#) = {},
Uargs(c_14) = {}, Uargs(tail^#) = {}, Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
plusIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
ifPlus(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
sum(x1) = [0] x1 + [0]
sumIter(x1, x2) = [0] x1 + [0] x2 + [0]
ifSum(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
isempty(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
error() = [0]
a() = [0]
b() = [0]
c() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plusIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
ifPlus^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
sumIter^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
ifSum^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
isempty^#(x1) = [0] x1 + [0]
c_11() = [0]
c_12() = [0]
head^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_15() = [0]
c_16(x1) = [0] x1 + [0]
a^#() = [0]
c_17() = [0]
c_18() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {a^#() -> c_18()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
a^#() = [7]
c_18() = [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.