Tool CaT
stdout:
MAYBE
Problem:
times(x,y) -> sum(generate(x,y))
generate(x,y) -> gen(x,y,0())
gen(x,y,z) -> if(ge(z,x),x,y,z)
if(true(),x,y,z) -> nil()
if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
sum(nil()) -> 0()
sum(cons(0(),xs)) -> sum(xs)
sum(cons(s(x),xs)) -> s(sum(cons(x,xs)))
ge(x,0()) -> true()
ge(0(),s(y)) -> false()
ge(s(x),s(y)) -> ge(x,y)
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:
{ times(x, y) -> sum(generate(x, y))
, generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, sum(nil()) -> 0()
, sum(cons(0(), xs)) -> sum(xs)
, sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
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: times^#(x, y) -> c_0(sum^#(generate(x, y)))
, 2: generate^#(x, y) -> c_1(gen^#(x, y, 0()))
, 3: gen^#(x, y, z) -> c_2(if^#(ge(z, x), x, y, z))
, 4: if^#(true(), x, y, z) -> c_3()
, 5: if^#(false(), x, y, z) -> c_4(gen^#(x, y, s(z)))
, 6: sum^#(nil()) -> c_5()
, 7: sum^#(cons(0(), xs)) -> c_6(sum^#(xs))
, 8: sum^#(cons(s(x), xs)) -> c_7(sum^#(cons(x, xs)))
, 9: ge^#(x, 0()) -> c_8()
, 10: ge^#(0(), s(y)) -> c_9()
, 11: ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
|->{9} [ YES(?,O(n^3)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{2} [ inherited ]
|
`->{3,5} [ NA ]
|
`->{4} [ NA ]
->{1} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{8,7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ times^#(x, y) -> c_0(sum^#(generate(x, y)))
, sum^#(nil()) -> c_5()
, generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{8,7}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{8,7}->{6}: NA
------------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3,5}.
* Path {2}->{3,5}: NA
-------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3,5}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {1},
Uargs(gen^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
gen(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [3]
true() = [1]
[0]
[1]
nil() = [0]
[0]
[0]
false() = [1]
[0]
[1]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
gen^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
if^#(x1, x2, x3, x4) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
gen(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
nil() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gen^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
ge^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_10(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {11}->{9}: YES(?,O(n^3))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
gen(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
nil() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gen^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(x, 0()) -> c_8()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
ge^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_8() = [1]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {11}->{10}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
gen(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
nil() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gen^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(y)) -> c_9()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
ge^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_9() = [1]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: times^#(x, y) -> c_0(sum^#(generate(x, y)))
, 2: generate^#(x, y) -> c_1(gen^#(x, y, 0()))
, 3: gen^#(x, y, z) -> c_2(if^#(ge(z, x), x, y, z))
, 4: if^#(true(), x, y, z) -> c_3()
, 5: if^#(false(), x, y, z) -> c_4(gen^#(x, y, s(z)))
, 6: sum^#(nil()) -> c_5()
, 7: sum^#(cons(0(), xs)) -> c_6(sum^#(xs))
, 8: sum^#(cons(s(x), xs)) -> c_7(sum^#(cons(x, xs)))
, 9: ge^#(x, 0()) -> c_8()
, 10: ge^#(0(), s(y)) -> c_9()
, 11: ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^2)) ]
|
|->{9} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{2} [ inherited ]
|
`->{3,5} [ NA ]
|
`->{4} [ NA ]
->{1} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{8,7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ times^#(x, y) -> c_0(sum^#(generate(x, y)))
, sum^#(nil()) -> c_5()
, generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{8,7}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{8,7}->{6}: NA
------------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3,5}.
* Path {2}->{3,5}: NA
-------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3,5}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {1},
Uargs(gen^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
generate(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gen(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [3]
true() = [1]
[1]
nil() = [0]
[0]
false() = [1]
[1]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
generate^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
gen^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3, x4) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: 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(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
generate(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gen(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
nil() = [0]
[0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
generate^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
gen^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
ge^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_10(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {11}->{9}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
generate(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gen(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
nil() = [0]
[0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
generate^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
gen^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(x, 0()) -> c_8()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
ge^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_8() = [1]
[0]
c_10(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {11}->{10}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
generate(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gen(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
nil() = [0]
[0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
generate^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
gen^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(y)) -> c_9()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
ge^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [7]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: times^#(x, y) -> c_0(sum^#(generate(x, y)))
, 2: generate^#(x, y) -> c_1(gen^#(x, y, 0()))
, 3: gen^#(x, y, z) -> c_2(if^#(ge(z, x), x, y, z))
, 4: if^#(true(), x, y, z) -> c_3()
, 5: if^#(false(), x, y, z) -> c_4(gen^#(x, y, s(z)))
, 6: sum^#(nil()) -> c_5()
, 7: sum^#(cons(0(), xs)) -> c_6(sum^#(xs))
, 8: sum^#(cons(s(x), xs)) -> c_7(sum^#(cons(x, xs)))
, 9: ge^#(x, 0()) -> c_8()
, 10: ge^#(0(), s(y)) -> c_9()
, 11: ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
|->{9} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{2} [ inherited ]
|
`->{3,5} [ NA ]
|
`->{4} [ NA ]
->{1} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{8,7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ times^#(x, y) -> c_0(sum^#(generate(x, y)))
, sum^#(nil()) -> c_5()
, generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{8,7}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{8,7}->{6}: NA
------------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3,5}.
* Path {2}->{3,5}: NA
-------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3,5}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {1},
Uargs(gen^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {1},
Uargs(c_4) = {1}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
generate(x1, x2) = [0] x1 + [0] x2 + [0]
gen(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [3]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [1] x1 + [1] x2 + [3]
true() = [1]
nil() = [0]
false() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [3]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
generate^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
gen^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [1] x1 + [0]
if^#(x1, x2, x3, x4) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: 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(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
generate(x1, x2) = [0] x1 + [0] x2 + [0]
gen(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
nil() = [0]
false() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
generate^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
gen^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
ge^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_10(x1) = [1] x1 + [7]
* Path {11}->{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(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
generate(x1, x2) = [0] x1 + [0] x2 + [0]
gen(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
nil() = [0]
false() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
generate^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
gen^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(x, 0()) -> c_8()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_8() = [1]
c_10(x1) = [1] x1 + [7]
* Path {11}->{10}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
generate(x1, x2) = [0] x1 + [0] x2 + [0]
gen(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
nil() = [0]
false() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
generate^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
gen^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(y)) -> c_9()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_9() = [1]
c_10(x1) = [1] x1 + [7]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ times(x, y) -> sum(generate(x, y))
, generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, sum(nil()) -> 0()
, sum(cons(0(), xs)) -> sum(xs)
, sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
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: times^#(x, y) -> c_0(sum^#(generate(x, y)))
, 2: generate^#(x, y) -> c_1(gen^#(x, y, 0()))
, 3: gen^#(x, y, z) -> c_2(if^#(ge(z, x), x, y, z))
, 4: if^#(true(), x, y, z) -> c_3()
, 5: if^#(false(), x, y, z) -> c_4(y, gen^#(x, y, s(z)))
, 6: sum^#(nil()) -> c_5()
, 7: sum^#(cons(0(), xs)) -> c_6(sum^#(xs))
, 8: sum^#(cons(s(x), xs)) -> c_7(sum^#(cons(x, xs)))
, 9: ge^#(x, 0()) -> c_8()
, 10: ge^#(0(), s(y)) -> c_9()
, 11: ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
|->{9} [ YES(?,O(n^3)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{2} [ inherited ]
|
`->{3,5} [ NA ]
|
`->{4} [ NA ]
->{1} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{8,7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ times^#(x, y) -> c_0(sum^#(generate(x, y)))
, sum^#(nil()) -> c_5()
, generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{8,7}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{8,7}->{6}: NA
------------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3,5}.
* Path {2}->{3,5}: NA
-------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3,5}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {1},
Uargs(gen^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {1},
Uargs(c_4) = {2}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
gen(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [3]
true() = [1]
[0]
[1]
nil() = [0]
[0]
[0]
false() = [1]
[0]
[1]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
gen^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
if^#(x1, x2, x3, x4) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* 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(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
gen(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
nil() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gen^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
ge^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_10(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {11}->{9}: YES(?,O(n^3))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
gen(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
nil() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gen^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(x, 0()) -> c_8()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
ge^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_8() = [1]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {11}->{10}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
sum(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
gen(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
nil() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
sum^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
generate^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gen^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(y)) -> c_9()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
ge^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_9() = [1]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: times^#(x, y) -> c_0(sum^#(generate(x, y)))
, 2: generate^#(x, y) -> c_1(gen^#(x, y, 0()))
, 3: gen^#(x, y, z) -> c_2(if^#(ge(z, x), x, y, z))
, 4: if^#(true(), x, y, z) -> c_3()
, 5: if^#(false(), x, y, z) -> c_4(y, gen^#(x, y, s(z)))
, 6: sum^#(nil()) -> c_5()
, 7: sum^#(cons(0(), xs)) -> c_6(sum^#(xs))
, 8: sum^#(cons(s(x), xs)) -> c_7(sum^#(cons(x, xs)))
, 9: ge^#(x, 0()) -> c_8()
, 10: ge^#(0(), s(y)) -> c_9()
, 11: ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^2)) ]
|
|->{9} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{2} [ inherited ]
|
`->{3,5} [ NA ]
|
`->{4} [ NA ]
->{1} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{8,7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ times^#(x, y) -> c_0(sum^#(generate(x, y)))
, sum^#(nil()) -> c_5()
, generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{8,7}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{8,7}->{6}: NA
------------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3,5}.
* Path {2}->{3,5}: NA
-------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3,5}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {1},
Uargs(gen^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {1},
Uargs(c_4) = {2}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
generate(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gen(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [3]
true() = [1]
[1]
nil() = [0]
[0]
false() = [1]
[1]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
generate^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
gen^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3, x4) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: 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(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
generate(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gen(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
nil() = [0]
[0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
generate^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
gen^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
ge^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_10(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {11}->{9}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
generate(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gen(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
nil() = [0]
[0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
generate^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
gen^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(x, 0()) -> c_8()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
ge^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_8() = [1]
[0]
c_10(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {11}->{10}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
generate(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gen(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
nil() = [0]
[0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
generate^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
gen^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(y)) -> c_9()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
ge^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [7]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: times^#(x, y) -> c_0(sum^#(generate(x, y)))
, 2: generate^#(x, y) -> c_1(gen^#(x, y, 0()))
, 3: gen^#(x, y, z) -> c_2(if^#(ge(z, x), x, y, z))
, 4: if^#(true(), x, y, z) -> c_3()
, 5: if^#(false(), x, y, z) -> c_4(y, gen^#(x, y, s(z)))
, 6: sum^#(nil()) -> c_5()
, 7: sum^#(cons(0(), xs)) -> c_6(sum^#(xs))
, 8: sum^#(cons(s(x), xs)) -> c_7(sum^#(cons(x, xs)))
, 9: ge^#(x, 0()) -> c_8()
, 10: ge^#(0(), s(y)) -> c_9()
, 11: ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
|->{9} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{2} [ inherited ]
|
`->{3,5} [ NA ]
|
`->{4} [ NA ]
->{1} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{8,7} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ times^#(x, y) -> c_0(sum^#(generate(x, y)))
, sum^#(nil()) -> c_5()
, generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1}->{8,7}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{8,7}->{6}.
* Path {1}->{8,7}->{6}: NA
------------------------
The usable rules for this path are:
{ generate(x, y) -> gen(x, y, 0())
, gen(x, y, z) -> if(ge(z, x), x, y, z)
, if(true(), x, y, z) -> nil()
, if(false(), x, y, z) -> cons(y, gen(x, y, s(z)))
, ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: inherited
-------------------
This path is subsumed by the proof of path {2}->{3,5}.
* Path {2}->{3,5}: NA
-------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}->{3,5}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(x, 0()) -> true()
, ge(0(), s(y)) -> false()
, ge(s(x), s(y)) -> ge(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {1},
Uargs(gen^#) = {}, Uargs(c_2) = {1}, Uargs(if^#) = {1},
Uargs(c_4) = {2}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
generate(x1, x2) = [0] x1 + [0] x2 + [0]
gen(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [3]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [1] x1 + [1] x2 + [3]
true() = [1]
nil() = [0]
false() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [3]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
generate^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
gen^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [1] x1 + [0]
if^#(x1, x2, x3, x4) = [3] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [1] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {11}: 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(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
generate(x1, x2) = [0] x1 + [0] x2 + [0]
gen(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
nil() = [0]
false() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
generate^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
gen^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
ge^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_10(x1) = [1] x1 + [7]
* Path {11}->{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(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
generate(x1, x2) = [0] x1 + [0] x2 + [0]
gen(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
nil() = [0]
false() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
generate^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
gen^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(x, 0()) -> c_8()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_8() = [1]
c_10(x1) = [1] x1 + [7]
* Path {11}->{10}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(times) = {}, Uargs(sum) = {}, Uargs(generate) = {},
Uargs(gen) = {}, Uargs(if) = {}, Uargs(ge) = {}, Uargs(cons) = {},
Uargs(s) = {}, Uargs(times^#) = {}, Uargs(c_0) = {},
Uargs(sum^#) = {}, Uargs(generate^#) = {}, Uargs(c_1) = {},
Uargs(gen^#) = {}, Uargs(c_2) = {}, Uargs(if^#) = {},
Uargs(c_4) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
generate(x1, x2) = [0] x1 + [0] x2 + [0]
gen(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
nil() = [0]
false() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
generate^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
gen^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3() = [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(y)) -> c_9()}
Weak Rules: {ge^#(s(x), s(y)) -> c_10(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_9() = [1]
c_10(x1) = [1] x1 + [7]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.