Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ from(X) -> cons(X, from(s(X)))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, Z))
, 2ndsneg(0(), Z) -> rnil()
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, Z))
, pi(X) -> 2ndspos(X, from(0()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, square(X) -> times(X, X)}
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: from^#(X) -> c_0(from^#(s(X)))
, 2: 2ndspos^#(0(), Z) -> c_1()
, 3: 2ndspos^#(s(N), cons(X, Z)) ->
c_2(2ndspos^#(s(N), cons2(X, Z)))
, 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_3(2ndsneg^#(N, Z))
, 5: 2ndsneg^#(0(), Z) -> c_4()
, 6: 2ndsneg^#(s(N), cons(X, Z)) ->
c_5(2ndsneg^#(s(N), cons2(X, Z)))
, 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_6(2ndspos^#(N, Z))
, 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
, 9: plus^#(0(), Y) -> c_8()
, 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
, 11: times^#(0(), Y) -> c_10()
, 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
, 13: square^#(X) -> c_12(times^#(X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ inherited ]
|
|->{11} [ YES(?,O(1)) ]
|
`->{12} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{8} [ inherited ]
|
|->{2} [ NA ]
|
`->{3,7,6,4} [ inherited ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
Uargs(square^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 0] [0]
2ndspos(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
rnil() = [0]
[0]
[0]
cons2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rcons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
posrecip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndsneg(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
negrecip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pi(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
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]
square(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [3 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
2ndspos^#(x1, 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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndsneg^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pi^#(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(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_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
square^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{2}: NA
-----------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}: inherited
------------------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{3,7,6,4}->{2}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}->{5}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{11}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndspos(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
rnil() = [0]
[0]
[0]
cons2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rcons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
posrecip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndsneg(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
negrecip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pi(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
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]
square(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndspos^#(x1, 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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndsneg^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pi^#(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(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_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
square^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(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(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(0(), Y) -> c_10()}
Weak Rules: {square^#(X) -> c_12(times^#(X, X))}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [2 0 2] x1 + [0 0 0] x2 + [0]
[2 0 2] [0 0 0] [0]
[2 2 2] [0 0 0] [0]
c_10() = [1]
[0]
[0]
square^#(x1) = [7 7 7] x1 + [7]
[7 7 7] [7]
[7 7 7] [7]
c_12(x1) = [2 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{9}: NA
------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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 {13}->{12}->{10}: inherited
--------------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{10}->{9}: NA
------------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: from^#(X) -> c_0(from^#(s(X)))
, 2: 2ndspos^#(0(), Z) -> c_1()
, 3: 2ndspos^#(s(N), cons(X, Z)) ->
c_2(2ndspos^#(s(N), cons2(X, Z)))
, 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_3(2ndsneg^#(N, Z))
, 5: 2ndsneg^#(0(), Z) -> c_4()
, 6: 2ndsneg^#(s(N), cons(X, Z)) ->
c_5(2ndsneg^#(s(N), cons2(X, Z)))
, 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_6(2ndspos^#(N, Z))
, 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
, 9: plus^#(0(), Y) -> c_8()
, 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
, 11: times^#(0(), Y) -> c_10()
, 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
, 13: square^#(X) -> c_12(times^#(X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ inherited ]
|
|->{11} [ YES(?,O(1)) ]
|
`->{12} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{8} [ inherited ]
|
|->{2} [ NA ]
|
`->{3,7,6,4} [ inherited ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
Uargs(square^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
2ndspos(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rcons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
2ndspos^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
pi^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{2}: NA
-----------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}: inherited
------------------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{3,7,6,4}->{2}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}->{5}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{11}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndspos(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rcons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndspos^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
pi^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(0(), Y) -> c_10()}
Weak Rules: {square^#(X) -> c_12(times^#(X, X))}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [0 2] x1 + [0 0] x2 + [0]
[2 0] [0 0] [0]
c_10() = [1]
[0]
square^#(x1) = [7 7] x1 + [7]
[7 7] [7]
c_12(x1) = [2 0] x1 + [7]
[0 0] [7]
* Path {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{9}: NA
------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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 {13}->{12}->{10}: inherited
--------------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{10}->{9}: NA
------------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: from^#(X) -> c_0(from^#(s(X)))
, 2: 2ndspos^#(0(), Z) -> c_1()
, 3: 2ndspos^#(s(N), cons(X, Z)) ->
c_2(2ndspos^#(s(N), cons2(X, Z)))
, 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_3(2ndsneg^#(N, Z))
, 5: 2ndsneg^#(0(), Z) -> c_4()
, 6: 2ndsneg^#(s(N), cons(X, Z)) ->
c_5(2ndsneg^#(s(N), cons2(X, Z)))
, 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_6(2ndspos^#(N, Z))
, 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
, 9: plus^#(0(), Y) -> c_8()
, 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
, 11: times^#(0(), Y) -> c_10()
, 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
, 13: square^#(X) -> c_12(times^#(X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ inherited ]
|
|->{11} [ NA ]
|
`->{12} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{8} [ inherited ]
|
|->{2} [ NA ]
|
`->{3,7,6,4} [ inherited ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
Uargs(square^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
2ndspos(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
rnil() = [0]
cons2(x1, x2) = [0] x1 + [0] x2 + [0]
rcons(x1, x2) = [0] x1 + [0] x2 + [0]
posrecip(x1) = [0] x1 + [0]
2ndsneg(x1, x2) = [0] x1 + [0] x2 + [0]
negrecip(x1) = [0] x1 + [0]
pi(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
square(x1) = [0] x1 + [0]
from^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
2ndspos^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
pi^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{2}: NA
-----------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}: inherited
------------------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{3,7,6,4}->{2}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}->{5}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{11}: NA
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
2ndspos(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
rnil() = [0]
cons2(x1, x2) = [0] x1 + [0] x2 + [0]
rcons(x1, x2) = [0] x1 + [0] x2 + [0]
posrecip(x1) = [0] x1 + [0]
2ndsneg(x1, x2) = [0] x1 + [0] x2 + [0]
negrecip(x1) = [0] x1 + [0]
pi(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
square(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
2ndspos^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
pi^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{9}: NA
------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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 {13}->{12}->{10}: inherited
--------------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{10}->{9}: NA
------------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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.
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:
{ from(X) -> cons(X, from(s(X)))
, 2ndspos(0(), Z) -> rnil()
, 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
, 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
rcons(posrecip(Y), 2ndsneg(N, Z))
, 2ndsneg(0(), Z) -> rnil()
, 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
, 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
rcons(negrecip(Y), 2ndspos(N, Z))
, pi(X) -> 2ndspos(X, from(0()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, square(X) -> times(X, X)}
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: from^#(X) -> c_0(X, from^#(s(X)))
, 2: 2ndspos^#(0(), Z) -> c_1()
, 3: 2ndspos^#(s(N), cons(X, Z)) ->
c_2(2ndspos^#(s(N), cons2(X, Z)))
, 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) ->
c_3(Y, 2ndsneg^#(N, Z))
, 5: 2ndsneg^#(0(), Z) -> c_4()
, 6: 2ndsneg^#(s(N), cons(X, Z)) ->
c_5(2ndsneg^#(s(N), cons2(X, Z)))
, 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) ->
c_6(Y, 2ndspos^#(N, Z))
, 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
, 9: plus^#(0(), Y) -> c_8(Y)
, 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
, 11: times^#(0(), Y) -> c_10()
, 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
, 13: square^#(X) -> c_12(times^#(X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ inherited ]
|
|->{11} [ YES(?,O(1)) ]
|
`->{12} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{8} [ inherited ]
|
|->{2} [ NA ]
|
`->{3,7,6,4} [ inherited ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 1 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
2ndspos(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
rnil() = [0]
[0]
[0]
cons2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rcons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
posrecip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndsneg(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
negrecip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pi(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
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]
square(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [1 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
2ndspos^#(x1, 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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
2ndsneg^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
pi^#(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
square^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_0(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{2}: NA
-----------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}: inherited
------------------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{3,7,6,4}->{2}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}->{5}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{11}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndspos(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
rnil() = [0]
[0]
[0]
cons2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rcons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
posrecip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2ndsneg(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
negrecip(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pi(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
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]
square(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
2ndspos^#(x1, 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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
2ndsneg^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
pi^#(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
square^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(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(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(0(), Y) -> c_10()}
Weak Rules: {square^#(X) -> c_12(times^#(X, X))}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [2 0 2] x1 + [0 0 0] x2 + [0]
[2 0 2] [0 0 0] [0]
[2 2 2] [0 0 0] [0]
c_10() = [1]
[0]
[0]
square^#(x1) = [7 7 7] x1 + [7]
[7 7 7] [7]
[7 7 7] [7]
c_12(x1) = [2 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{9}: NA
------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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 {13}->{12}->{10}: inherited
--------------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{10}->{9}: NA
------------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: from^#(X) -> c_0(X, from^#(s(X)))
, 2: 2ndspos^#(0(), Z) -> c_1()
, 3: 2ndspos^#(s(N), cons(X, Z)) ->
c_2(2ndspos^#(s(N), cons2(X, Z)))
, 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) ->
c_3(Y, 2ndsneg^#(N, Z))
, 5: 2ndsneg^#(0(), Z) -> c_4()
, 6: 2ndsneg^#(s(N), cons(X, Z)) ->
c_5(2ndsneg^#(s(N), cons2(X, Z)))
, 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) ->
c_6(Y, 2ndspos^#(N, Z))
, 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
, 9: plus^#(0(), Y) -> c_8(Y)
, 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
, 11: times^#(0(), Y) -> c_10()
, 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
, 13: square^#(X) -> c_12(times^#(X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ inherited ]
|
|->{11} [ YES(?,O(1)) ]
|
`->{12} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{8} [ inherited ]
|
|->{2} [ NA ]
|
`->{3,7,6,4} [ inherited ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
2ndspos(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rcons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_0(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
2ndspos^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pi^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_0(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{2}: NA
-----------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}: inherited
------------------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{3,7,6,4}->{2}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}->{5}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{11}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndspos(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
rnil() = [0]
[0]
cons2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rcons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
posrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
2ndsneg(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
negrecip(x1) = [0 0] x1 + [0]
[0 0] [0]
pi(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
2ndspos^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
pi^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(0(), Y) -> c_10()}
Weak Rules: {square^#(X) -> c_12(times^#(X, X))}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [0 2] x1 + [0 0] x2 + [0]
[2 0] [0 0] [0]
c_10() = [1]
[0]
square^#(x1) = [7 7] x1 + [7]
[7 7] [7]
c_12(x1) = [2 0] x1 + [7]
[0 0] [7]
* Path {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{9}: NA
------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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 {13}->{12}->{10}: inherited
--------------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{10}->{9}: NA
------------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: from^#(X) -> c_0(X, from^#(s(X)))
, 2: 2ndspos^#(0(), Z) -> c_1()
, 3: 2ndspos^#(s(N), cons(X, Z)) ->
c_2(2ndspos^#(s(N), cons2(X, Z)))
, 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) ->
c_3(Y, 2ndsneg^#(N, Z))
, 5: 2ndsneg^#(0(), Z) -> c_4()
, 6: 2ndsneg^#(s(N), cons(X, Z)) ->
c_5(2ndsneg^#(s(N), cons2(X, Z)))
, 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) ->
c_6(Y, 2ndspos^#(N, Z))
, 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
, 9: plus^#(0(), Y) -> c_8(Y)
, 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
, 11: times^#(0(), Y) -> c_10()
, 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
, 13: square^#(X) -> c_12(times^#(X, X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ inherited ]
|
|->{11} [ NA ]
|
`->{12} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{8} [ inherited ]
|
|->{2} [ NA ]
|
`->{3,7,6,4} [ inherited ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
2ndspos(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
rnil() = [0]
cons2(x1, x2) = [0] x1 + [0] x2 + [0]
rcons(x1, x2) = [0] x1 + [0] x2 + [0]
posrecip(x1) = [0] x1 + [0]
2ndsneg(x1, x2) = [0] x1 + [0] x2 + [0]
negrecip(x1) = [0] x1 + [0]
pi(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
square(x1) = [0] x1 + [0]
from^#(x1) = [3] x1 + [0]
c_0(x1, x2) = [2] x1 + [1] x2 + [0]
2ndspos^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [0]
pi^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_0(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{2}: NA
-----------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}: inherited
------------------------------
This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.
* Path {8}->{3,7,6,4}->{2}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {8}->{3,7,6,4}->{5}: NA
----------------------------
The usable rules for this path are:
{from(X) -> cons(X, from(s(X)))}
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 {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{11}: NA
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
2ndspos(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
rnil() = [0]
cons2(x1, x2) = [0] x1 + [0] x2 + [0]
rcons(x1, x2) = [0] x1 + [0] x2 + [0]
posrecip(x1) = [0] x1 + [0]
2ndsneg(x1, x2) = [0] x1 + [0] x2 + [0]
negrecip(x1) = [0] x1 + [0]
pi(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
square(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
2ndspos^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [0]
pi^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_12(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{9}: NA
------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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 {13}->{12}->{10}: inherited
--------------------------------
This path is subsumed by the proof of path {13}->{12}->{10}->{9}.
* Path {13}->{12}->{10}->{9}: NA
------------------------------
The usable rules for this path are:
{ times(0(), Y) -> 0()
, times(s(X), Y) -> plus(Y, times(X, Y))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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.
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.