Tool CaT
stdout:
MAYBE
Problem:
log(x,s(s(y))) -> cond(le(x,s(s(y))),x,y)
cond(true(),x,y) -> s(0())
cond(false(),x,y) -> double(log(x,square(s(s(y)))))
le(0(),v) -> true()
le(s(u),0()) -> false()
le(s(u),s(v)) -> le(u,v)
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
square(0()) -> 0()
square(s(x)) -> s(plus(square(x),double(x)))
plus(n,0()) -> n
plus(n,s(m)) -> s(plus(n,m))
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:
{ log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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: log^#(x, s(s(y))) -> c_0(cond^#(le(x, s(s(y))), x, y))
, 2: cond^#(true(), x, y) -> c_1()
, 3: cond^#(false(), x, y) ->
c_2(double^#(log(x, square(s(s(y))))))
, 4: le^#(0(), v) -> c_3()
, 5: le^#(s(u), 0()) -> c_4()
, 6: le^#(s(u), s(v)) -> c_5(le^#(u, v))
, 7: double^#(0()) -> c_6()
, 8: double^#(s(x)) -> c_7(double^#(x))
, 9: square^#(0()) -> c_8()
, 10: square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
, 11: plus^#(n, 0()) -> c_10()
, 12: plus^#(n, s(m)) -> c_11(plus^#(n, m))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
|->{11} [ MAYBE ]
|
`->{12} [ inherited ]
|
`->{11} [ NA ]
->{9} [ YES(?,O(1)) ]
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
|->{2} [ NA ]
|
`->{3} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [3]
true() = [0]
[1]
0() = [0]
[0]
false() = [0]
[1]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
cond^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{3}->{7}: NA
----------------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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 {1}->{3}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{3}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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 {6}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_5(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {6}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(0(), v) -> c_3()}
Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 1] [2]
0() = [2]
[2]
le^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_3() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(u), 0()) -> c_4()}
Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [2]
0() = [0]
[2]
le^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(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]
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: {square^#(0()) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(square^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
square^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_8() = [0]
[1]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{12}->{11}.
* Path {10}->{11}: MAYBE
----------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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:
{ square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
, plus^#(n, 0()) -> c_10()
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
Proof Output:
The input cannot be shown compatible
* Path {10}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {10}->{12}->{11}.
* Path {10}->{12}->{11}: NA
-------------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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: log^#(x, s(s(y))) -> c_0(cond^#(le(x, s(s(y))), x, y))
, 2: cond^#(true(), x, y) -> c_1()
, 3: cond^#(false(), x, y) ->
c_2(double^#(log(x, square(s(s(y))))))
, 4: le^#(0(), v) -> c_3()
, 5: le^#(s(u), 0()) -> c_4()
, 6: le^#(s(u), s(v)) -> c_5(le^#(u, v))
, 7: double^#(0()) -> c_6()
, 8: double^#(s(x)) -> c_7(double^#(x))
, 9: square^#(0()) -> c_8()
, 10: square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
, 11: plus^#(n, 0()) -> c_10()
, 12: plus^#(n, s(m)) -> c_11(plus^#(n, m))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
|->{11} [ MAYBE ]
|
`->{12} [ inherited ]
|
`->{11} [ NA ]
->{9} [ YES(?,O(1)) ]
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
|->{2} [ NA ]
|
`->{3} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [2]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [2] x2 + [2]
true() = [1]
0() = [0]
false() = [1]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
cond^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(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 {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{3}->{7}: NA
----------------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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 {1}->{3}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{3}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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 {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(0(), v) -> c_3()}
Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
le^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_3() = [1]
c_5(x1) = [1] x1 + [5]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(u), 0()) -> c_4()}
Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [1]
c_5(x1) = [1] x1 + [7]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {square^#(0()) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(square^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
square^#(x1) = [1] x1 + [7]
c_8() = [1]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{12}->{11}.
* Path {10}->{11}: MAYBE
----------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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:
{ square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
, plus^#(n, 0()) -> c_10()
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
Proof Output:
The input cannot be shown compatible
* Path {10}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {10}->{12}->{11}.
* Path {10}->{12}->{11}: NA
-------------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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: log^#(x, s(s(y))) -> c_0(cond^#(le(x, s(s(y))), x, y))
, 2: cond^#(true(), x, y) -> c_1()
, 3: cond^#(false(), x, y) ->
c_2(double^#(log(x, square(s(s(y))))))
, 4: le^#(0(), v) -> c_3()
, 5: le^#(s(u), 0()) -> c_4()
, 6: le^#(s(u), s(v)) -> c_5(le^#(u, v))
, 7: double^#(0()) -> c_6()
, 8: double^#(s(x)) -> c_7(double^#(x))
, 9: square^#(0()) -> c_8()
, 10: square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
, 11: plus^#(n, 0()) -> c_10(n)
, 12: plus^#(n, s(m)) -> c_11(plus^#(n, m))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
|->{11} [ MAYBE ]
|
`->{12} [ inherited ]
|
`->{11} [ NA ]
->{9} [ YES(?,O(1)) ]
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
|->{2} [ NA ]
|
`->{3} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [3]
true() = [0]
[1]
0() = [0]
[0]
false() = [0]
[1]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
cond^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(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 {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{3}->{7}: NA
----------------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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 {1}->{3}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{3}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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 {6}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_5(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {6}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(0(), v) -> c_3()}
Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 1] [2]
0() = [2]
[2]
le^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_3() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(u), 0()) -> c_4()}
Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [2]
0() = [0]
[2]
le^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
square(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
square^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {square^#(0()) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(square^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
square^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_8() = [0]
[1]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{12}->{11}.
* Path {10}->{11}: MAYBE
----------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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:
{ square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
, plus^#(n, 0()) -> c_10(n)
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
Proof Output:
The input cannot be shown compatible
* Path {10}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {10}->{12}->{11}.
* Path {10}->{12}->{11}: NA
-------------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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: log^#(x, s(s(y))) -> c_0(cond^#(le(x, s(s(y))), x, y))
, 2: cond^#(true(), x, y) -> c_1()
, 3: cond^#(false(), x, y) ->
c_2(double^#(log(x, square(s(s(y))))))
, 4: le^#(0(), v) -> c_3()
, 5: le^#(s(u), 0()) -> c_4()
, 6: le^#(s(u), s(v)) -> c_5(le^#(u, v))
, 7: double^#(0()) -> c_6()
, 8: double^#(s(x)) -> c_7(double^#(x))
, 9: square^#(0()) -> c_8()
, 10: square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
, 11: plus^#(n, 0()) -> c_10(n)
, 12: plus^#(n, s(m)) -> c_11(plus^#(n, m))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
|->{11} [ MAYBE ]
|
`->{12} [ inherited ]
|
`->{11} [ NA ]
->{9} [ YES(?,O(1)) ]
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
|->{2} [ NA ]
|
`->{3} [ inherited ]
|
|->{7} [ NA ]
|
`->{8} [ inherited ]
|
`->{7} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{2}: NA
-----------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [2]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [2] x2 + [2]
true() = [1]
0() = [0]
false() = [1]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
cond^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(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 {1}->{3}: inherited
------------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{3}->{7}: NA
----------------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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 {1}->{3}->{8}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
* Path {1}->{3}->{8}->{7}: NA
---------------------------
The usable rules for this path are:
{ le(0(), v) -> true()
, le(s(u), 0()) -> false()
, le(s(u), s(v)) -> le(u, v)
, log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, cond(true(), x, y) -> s(0())
, cond(false(), x, y) -> double(log(x, square(s(s(y)))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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 {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(0(), v) -> c_3()}
Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
le^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_3() = [1]
c_5(x1) = [1] x1 + [5]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(u), 0()) -> c_4()}
Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [1]
c_5(x1) = [1] x1 + [7]
* Path {9}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
log(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
double(x1) = [0] x1 + [0]
square(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
log^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
square^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {square^#(0()) -> c_8()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(square^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
square^#(x1) = [1] x1 + [7]
c_8() = [1]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{12}->{11}.
* Path {10}->{11}: MAYBE
----------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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:
{ square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
, plus^#(n, 0()) -> c_10(n)
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
Proof Output:
The input cannot be shown compatible
* Path {10}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {10}->{12}->{11}.
* Path {10}->{12}->{11}: NA
-------------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, square(0()) -> 0()
, square(s(x)) -> s(plus(square(x), double(x)))
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))}
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) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.