Tool CaT
stdout:
MAYBE
Problem:
isEmpty(cons(x,xs)) -> false()
isEmpty(nil()) -> true()
isZero(0()) -> true()
isZero(s(x)) -> false()
head(cons(x,xs)) -> x
tail(cons(x,xs)) -> xs
tail(nil()) -> nil()
p(s(s(x))) -> s(p(s(x)))
p(s(0())) -> 0()
p(0()) -> 0()
inc(s(x)) -> s(inc(x))
inc(0()) -> s(0())
sumList(xs,y) -> if(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y))
if(true(),b,y,xs,ys,x) -> y
if(false(),true(),y,xs,ys,x) -> sumList(xs,y)
if(false(),false(),y,xs,ys,x) -> sumList(ys,x)
sum(xs) -> sumList(xs,0())
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:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())
, sumList(xs, y) ->
if(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y))
, if(true(), b, y, xs, ys, x) -> y
, if(false(), true(), y, xs, ys, x) -> sumList(xs, y)
, if(false(), false(), y, xs, ys, x) -> sumList(ys, x)
, sum(xs) -> sumList(xs, 0())}
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: isEmpty^#(cons(x, xs)) -> c_0()
, 2: isEmpty^#(nil()) -> c_1()
, 3: isZero^#(0()) -> c_2()
, 4: isZero^#(s(x)) -> c_3()
, 5: head^#(cons(x, xs)) -> c_4()
, 6: tail^#(cons(x, xs)) -> c_5()
, 7: tail^#(nil()) -> c_6()
, 8: p^#(s(s(x))) -> c_7(p^#(s(x)))
, 9: p^#(s(0())) -> c_8()
, 10: p^#(0()) -> c_9()
, 11: inc^#(s(x)) -> c_10(inc^#(x))
, 12: inc^#(0()) -> c_11()
, 13: sumList^#(xs, y) ->
c_12(if^#(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y)))
, 14: if^#(true(), b, y, xs, ys, x) -> c_13()
, 15: if^#(false(), true(), y, xs, ys, x) -> c_14(sumList^#(xs, y))
, 16: if^#(false(), false(), y, xs, ys, x) ->
c_15(sumList^#(ys, x))
, 17: sum^#(xs) -> c_16(sumList^#(xs, 0()))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{17} [ inherited ]
|
`->{13,16,15} [ MAYBE ]
|
`->{14} [ NA ]
->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(n^1)) ]
->{10} [ YES(?,O(1)) ]
->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isEmpty^#(cons(x, xs)) -> c_0()}
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_0() = [0]
[1]
* Path {2}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isEmpty^#(nil()) -> c_1()}
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_1() = [0]
[1]
* Path {3}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isZero^#(0()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isZero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
isZero^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isZero^#(s(x)) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isZero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [2]
[0 0] [2]
isZero^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {head^#(cons(x, xs)) -> c_4()}
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_4() = [0]
[1]
* Path {6}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(x, xs)) -> c_5()}
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_5() = [0]
[1]
* Path {7}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(nil()) -> c_6()}
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_6() = [0]
[1]
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {1}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [1]
[0 0] [0]
p^#(x1) = [1 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {8}->{9}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {1}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(0())) -> c_8()}
Weak Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [0 2] x1 + [2]
[0 1] [0]
p^#(x1) = [2 0] x1 + [0]
[2 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [3]
c_8() = [1]
[0]
* Path {10}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_9() = [0]
[1]
* Path {11}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {inc^#(s(x)) -> c_10(inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
inc^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11}->{12}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {inc^#(0()) -> c_11()}
Weak Rules: {inc^#(s(x)) -> c_10(inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
inc^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_10(x1) = [1 0] x1 + [5]
[2 0] [3]
c_11() = [1]
[0]
* Path {17}: inherited
--------------------
This path is subsumed by the proof of path {17}->{13,16,15}.
* Path {17}->{13,16,15}: MAYBE
----------------------------
The usable rules for this path are:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
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:
{ sum^#(xs) -> c_16(sumList^#(xs, 0()))
, sumList^#(xs, y) ->
c_12(if^#(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y)))
, if^#(false(), false(), y, xs, ys, x) -> c_15(sumList^#(ys, x))
, if^#(false(), true(), y, xs, ys, x) -> c_14(sumList^#(xs, y))
, isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
Proof Output:
The input cannot be shown compatible
* Path {17}->{13,16,15}->{14}: NA
-------------------------------
The usable rules for this path are:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {1, 2}, Uargs(isZero) = {1},
Uargs(s) = {1}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {1},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {1},
Uargs(if^#) = {1, 2, 4, 5, 6}, Uargs(c_14) = {1},
Uargs(c_15) = {1}, Uargs(sum^#) = {}, Uargs(c_16) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [3]
[3 3] [3]
cons(x1, x2) = [1 1] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [1 0] x1 + [3]
[3 3] [3]
0() = [1]
[2]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
head(x1) = [3 3] x1 + [3]
[3 3] [3]
tail(x1) = [2 0] x1 + [2]
[0 2] [0]
p(x1) = [2 0] x1 + [0]
[2 0] [0]
inc(x1) = [3 0] x1 + [0]
[0 2] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3, x4, x5, x6) = [3 0] x1 + [3 0] x2 + [0 0] x3 + [3 0] x4 + [3 0] x5 + [3 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: isEmpty^#(cons(x, xs)) -> c_0()
, 2: isEmpty^#(nil()) -> c_1()
, 3: isZero^#(0()) -> c_2()
, 4: isZero^#(s(x)) -> c_3()
, 5: head^#(cons(x, xs)) -> c_4()
, 6: tail^#(cons(x, xs)) -> c_5()
, 7: tail^#(nil()) -> c_6()
, 8: p^#(s(s(x))) -> c_7(p^#(s(x)))
, 9: p^#(s(0())) -> c_8()
, 10: p^#(0()) -> c_9()
, 11: inc^#(s(x)) -> c_10(inc^#(x))
, 12: inc^#(0()) -> c_11()
, 13: sumList^#(xs, y) ->
c_12(if^#(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y)))
, 14: if^#(true(), b, y, xs, ys, x) -> c_13()
, 15: if^#(false(), true(), y, xs, ys, x) -> c_14(sumList^#(xs, y))
, 16: if^#(false(), false(), y, xs, ys, x) ->
c_15(sumList^#(ys, x))
, 17: sum^#(xs) -> c_16(sumList^#(xs, 0()))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{17} [ inherited ]
|
`->{13,16,15} [ MAYBE ]
|
`->{14} [ NA ]
->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(n^1)) ]
->{10} [ YES(?,O(1)) ]
->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isEmpty^#(cons(x, xs)) -> c_0()}
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_0() = [1]
* Path {2}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isEmpty^#(nil()) -> c_1()}
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_1() = [1]
* Path {3}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isZero^#(0()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isZero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
isZero^#(x1) = [1] x1 + [7]
c_2() = [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isZero^#(s(x)) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isZero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [7]
isZero^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {head^#(cons(x, xs)) -> c_4()}
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_4() = [1]
* Path {6}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(x, xs)) -> c_5()}
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_5() = [1]
* Path {7}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(nil()) -> c_6()}
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_6() = [1]
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {1}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [3] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
p^#(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [3]
* Path {8}->{9}: YES(?,O(1))
--------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {1}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(0())) -> c_8()}
Weak Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [3]
p^#(x1) = [2] x1 + [2]
c_7(x1) = [1] x1 + [0]
c_8() = [1]
* Path {10}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_9() = [1]
* Path {11}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [3] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {inc^#(s(x)) -> c_10(inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
inc^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [7]
* Path {11}->{12}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1},
Uargs(sumList^#) = {}, Uargs(c_12) = {}, Uargs(if^#) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {inc^#(0()) -> c_11()}
Weak Rules: {inc^#(s(x)) -> c_10(inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
inc^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11() = [1]
* Path {17}: inherited
--------------------
This path is subsumed by the proof of path {17}->{13,16,15}.
* Path {17}->{13,16,15}: MAYBE
----------------------------
The usable rules for this path are:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
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:
{ sum^#(xs) -> c_16(sumList^#(xs, 0()))
, sumList^#(xs, y) ->
c_12(if^#(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y)))
, if^#(false(), false(), y, xs, ys, x) -> c_15(sumList^#(ys, x))
, if^#(false(), true(), y, xs, ys, x) -> c_14(sumList^#(xs, y))
, isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
Proof Output:
The input cannot be shown compatible
* Path {17}->{13,16,15}->{14}: NA
-------------------------------
The usable rules for this path are:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {1, 2}, Uargs(isZero) = {1},
Uargs(s) = {1}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {1},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(tail^#) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {}, Uargs(c_12) = {1},
Uargs(if^#) = {1, 2, 4, 5, 6}, Uargs(c_14) = {1},
Uargs(c_15) = {1}, Uargs(sum^#) = {}, Uargs(c_16) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [3] x1 + [3]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
false() = [0]
nil() = [1]
true() = [1]
isZero(x1) = [1] x1 + [0]
0() = [2]
s(x1) = [1] x1 + [1]
head(x1) = [3] x1 + [3]
tail(x1) = [3] x1 + [3]
p(x1) = [2] x1 + [0]
inc(x1) = [2] x1 + [2]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
tail^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [1] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [3] x1 + [3] x2 + [0] x3 + [3] x4 + [3] x5 + [3] x6 + [0]
c_13() = [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) '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:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())
, sumList(xs, y) ->
if(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y))
, if(true(), b, y, xs, ys, x) -> y
, if(false(), true(), y, xs, ys, x) -> sumList(xs, y)
, if(false(), false(), y, xs, ys, x) -> sumList(ys, x)
, sum(xs) -> sumList(xs, 0())}
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: isEmpty^#(cons(x, xs)) -> c_0()
, 2: isEmpty^#(nil()) -> c_1()
, 3: isZero^#(0()) -> c_2()
, 4: isZero^#(s(x)) -> c_3()
, 5: head^#(cons(x, xs)) -> c_4(x)
, 6: tail^#(cons(x, xs)) -> c_5(xs)
, 7: tail^#(nil()) -> c_6()
, 8: p^#(s(s(x))) -> c_7(p^#(s(x)))
, 9: p^#(s(0())) -> c_8()
, 10: p^#(0()) -> c_9()
, 11: inc^#(s(x)) -> c_10(inc^#(x))
, 12: inc^#(0()) -> c_11()
, 13: sumList^#(xs, y) ->
c_12(if^#(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y)))
, 14: if^#(true(), b, y, xs, ys, x) -> c_13(y)
, 15: if^#(false(), true(), y, xs, ys, x) -> c_14(sumList^#(xs, y))
, 16: if^#(false(), false(), y, xs, ys, x) ->
c_15(sumList^#(ys, x))
, 17: sum^#(xs) -> c_16(sumList^#(xs, 0()))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{17} [ inherited ]
|
`->{13,16,15} [ MAYBE ]
|
`->{14} [ NA ]
->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(n^1)) ]
->{10} [ YES(?,O(1)) ]
->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(n^2)) ]
->{5} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isEmpty^#(cons(x, xs)) -> c_0()}
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_0() = [0]
[1]
* Path {2}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isEmpty^#(nil()) -> c_1()}
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_1() = [0]
[1]
* Path {3}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isZero^#(0()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isZero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
isZero^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isZero^#(s(x)) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isZero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [2]
[0 0] [2]
isZero^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(cons(x, xs)) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_4) = {}
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_4(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(x, xs)) -> c_5(xs)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_5) = {}
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_5(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {7}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(nil()) -> c_6()}
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_6() = [0]
[1]
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [1]
[0 0] [0]
p^#(x1) = [1 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {8}->{9}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(0())) -> c_8()}
Weak Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [0 2] x1 + [2]
[0 1] [0]
p^#(x1) = [2 0] x1 + [0]
[2 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [3]
c_8() = [1]
[0]
* Path {10}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_9() = [0]
[1]
* Path {11}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {1}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {inc^#(s(x)) -> c_10(inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
inc^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11}->{12}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {1}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
true() = [0]
[0]
isZero(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
inc(x1) = [0 0] x1 + [0]
[0 0] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {inc^#(0()) -> c_11()}
Weak Rules: {inc^#(s(x)) -> c_10(inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
inc^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_10(x1) = [1 0] x1 + [5]
[2 0] [3]
c_11() = [1]
[0]
* Path {17}: inherited
--------------------
This path is subsumed by the proof of path {17}->{13,16,15}.
* Path {17}->{13,16,15}: MAYBE
----------------------------
The usable rules for this path are:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
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:
{ sum^#(xs) -> c_16(sumList^#(xs, 0()))
, sumList^#(xs, y) ->
c_12(if^#(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y)))
, if^#(false(), false(), y, xs, ys, x) -> c_15(sumList^#(ys, x))
, if^#(false(), true(), y, xs, ys, x) -> c_14(sumList^#(xs, y))
, isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
Proof Output:
The input cannot be shown compatible
* Path {17}->{13,16,15}->{14}: NA
-------------------------------
The usable rules for this path are:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {1}, Uargs(cons) = {1, 2}, Uargs(isZero) = {1},
Uargs(s) = {1}, Uargs(head) = {1}, Uargs(tail) = {1},
Uargs(p) = {1}, Uargs(inc) = {1}, Uargs(sumList) = {},
Uargs(if) = {}, Uargs(sum) = {}, Uargs(isEmpty^#) = {},
Uargs(isZero^#) = {}, Uargs(head^#) = {}, Uargs(c_4) = {},
Uargs(tail^#) = {}, Uargs(c_5) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {1, 2}, Uargs(c_12) = {1},
Uargs(if^#) = {1, 2, 3, 4, 5, 6}, Uargs(c_13) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(sum^#) = {},
Uargs(c_16) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [2 0] x1 + [0]
[3 3] [3]
cons(x1, x2) = [1 1] x1 + [1 3] x2 + [2]
[0 1] [0 1] [0]
false() = [1]
[1]
nil() = [2]
[0]
true() = [1]
[1]
isZero(x1) = [2 0] x1 + [0]
[1 3] [3]
0() = [3]
[1]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
head(x1) = [3 3] x1 + [3]
[3 3] [3]
tail(x1) = [2 0] x1 + [0]
[0 2] [0]
p(x1) = [2 0] x1 + [0]
[3 0] [0]
inc(x1) = [2 0] x1 + [0]
[2 2] [0]
sumList(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4, x5, x6) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
isEmpty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
isZero^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
inc^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11() = [0]
[0]
sumList^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3, x4, x5, x6) = [3 0] x1 + [3 0] x2 + [3 0] x3 + [3 0] x4 + [3 0] x5 + [3 3] x6 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: isEmpty^#(cons(x, xs)) -> c_0()
, 2: isEmpty^#(nil()) -> c_1()
, 3: isZero^#(0()) -> c_2()
, 4: isZero^#(s(x)) -> c_3()
, 5: head^#(cons(x, xs)) -> c_4(x)
, 6: tail^#(cons(x, xs)) -> c_5(xs)
, 7: tail^#(nil()) -> c_6()
, 8: p^#(s(s(x))) -> c_7(p^#(s(x)))
, 9: p^#(s(0())) -> c_8()
, 10: p^#(0()) -> c_9()
, 11: inc^#(s(x)) -> c_10(inc^#(x))
, 12: inc^#(0()) -> c_11()
, 13: sumList^#(xs, y) ->
c_12(if^#(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y)))
, 14: if^#(true(), b, y, xs, ys, x) -> c_13(y)
, 15: if^#(false(), true(), y, xs, ys, x) -> c_14(sumList^#(xs, y))
, 16: if^#(false(), false(), y, xs, ys, x) ->
c_15(sumList^#(ys, x))
, 17: sum^#(xs) -> c_16(sumList^#(xs, 0()))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{17} [ inherited ]
|
`->{13,16,15} [ MAYBE ]
|
`->{14} [ NA ]
->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(n^1)) ]
->{10} [ YES(?,O(1)) ]
->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(1)) ]
->{7} [ YES(?,O(1)) ]
->{6} [ YES(?,O(n^1)) ]
->{5} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isEmpty^#(cons(x, xs)) -> c_0()}
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_0() = [1]
* Path {2}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isEmpty^#(nil()) -> c_1()}
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_1() = [1]
* Path {3}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isZero^#(0()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isZero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
isZero^#(x1) = [1] x1 + [7]
c_2() = [1]
* Path {4}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isZero^#(s(x)) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(isZero^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [7]
isZero^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [3] x1 + [0]
c_4(x1) = [1] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(cons(x, xs)) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_4(x1) = [1] x1 + [1]
* Path {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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [3] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(x, xs)) -> c_5(xs)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_5(x1) = [1] x1 + [1]
* Path {7}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(nil()) -> c_6()}
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_6() = [1]
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [3] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
p^#(x1) = [1] x1 + [0]
c_7(x1) = [1] x1 + [3]
* Path {8}->{9}: YES(?,O(1))
--------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(0())) -> c_8()}
Weak Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [3]
p^#(x1) = [2] x1 + [2]
c_7(x1) = [1] x1 + [0]
c_8() = [1]
* Path {10}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_9()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_9() = [1]
* Path {11}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {1}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [3] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {inc^#(s(x)) -> c_10(inc^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
inc^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [7]
* Path {11}->{12}: 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(isEmpty) = {}, Uargs(cons) = {}, Uargs(isZero) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(tail) = {}, Uargs(p) = {},
Uargs(inc) = {}, Uargs(sumList) = {}, Uargs(if) = {},
Uargs(sum) = {}, Uargs(isEmpty^#) = {}, Uargs(isZero^#) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(tail^#) = {},
Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
Uargs(inc^#) = {}, Uargs(c_10) = {1}, Uargs(sumList^#) = {},
Uargs(c_12) = {}, Uargs(if^#) = {}, Uargs(c_13) = {},
Uargs(c_14) = {}, Uargs(c_15) = {}, Uargs(sum^#) = {},
Uargs(c_16) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
nil() = [0]
true() = [0]
isZero(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
inc(x1) = [0] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {inc^#(0()) -> c_11()}
Weak Rules: {inc^#(s(x)) -> c_10(inc^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
inc^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [0]
c_11() = [1]
* Path {17}: inherited
--------------------
This path is subsumed by the proof of path {17}->{13,16,15}.
* Path {17}->{13,16,15}: MAYBE
----------------------------
The usable rules for this path are:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
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:
{ sum^#(xs) -> c_16(sumList^#(xs, 0()))
, sumList^#(xs, y) ->
c_12(if^#(isEmpty(xs),
isZero(head(xs)),
y,
tail(xs),
cons(p(head(xs)), tail(xs)),
inc(y)))
, if^#(false(), false(), y, xs, ys, x) -> c_15(sumList^#(ys, x))
, if^#(false(), true(), y, xs, ys, x) -> c_14(sumList^#(xs, y))
, isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
Proof Output:
The input cannot be shown compatible
* Path {17}->{13,16,15}->{14}: NA
-------------------------------
The usable rules for this path are:
{ isEmpty(cons(x, xs)) -> false()
, isEmpty(nil()) -> true()
, isZero(0()) -> true()
, isZero(s(x)) -> false()
, head(cons(x, xs)) -> x
, tail(cons(x, xs)) -> xs
, tail(nil()) -> nil()
, p(s(s(x))) -> s(p(s(x)))
, p(s(0())) -> 0()
, p(0()) -> 0()
, inc(s(x)) -> s(inc(x))
, inc(0()) -> s(0())}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isEmpty) = {1}, Uargs(cons) = {1, 2}, Uargs(isZero) = {1},
Uargs(s) = {1}, Uargs(head) = {1}, Uargs(tail) = {1},
Uargs(p) = {1}, Uargs(inc) = {1}, Uargs(sumList) = {},
Uargs(if) = {}, Uargs(sum) = {}, Uargs(isEmpty^#) = {},
Uargs(isZero^#) = {}, Uargs(head^#) = {}, Uargs(c_4) = {},
Uargs(tail^#) = {}, Uargs(c_5) = {}, Uargs(p^#) = {},
Uargs(c_7) = {}, Uargs(inc^#) = {}, Uargs(c_10) = {},
Uargs(sumList^#) = {1, 2}, Uargs(c_12) = {1},
Uargs(if^#) = {1, 2, 3, 4, 5, 6}, Uargs(c_13) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(sum^#) = {},
Uargs(c_16) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isEmpty(x1) = [1] x1 + [3]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
false() = [1]
nil() = [3]
true() = [1]
isZero(x1) = [2] x1 + [0]
0() = [2]
s(x1) = [1] x1 + [2]
head(x1) = [1] x1 + [3]
tail(x1) = [1] x1 + [3]
p(x1) = [2] x1 + [0]
inc(x1) = [3] x1 + [0]
sumList(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4, x5, x6) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0] x6 + [0]
sum(x1) = [0] x1 + [0]
isEmpty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
isZero^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
p^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8() = [0]
c_9() = [0]
inc^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11() = [0]
sumList^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_12(x1) = [1] x1 + [0]
if^#(x1, x2, x3, x4, x5, x6) = [3] x1 + [3] x2 + [3] x3 + [3] x4 + [3] x5 + [3] x6 + [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_16(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.