Tool CaT
stdout:
MAYBE
Problem:
1024() -> 1024_1(0())
1024_1(x) -> if(lt(x,10()),x)
if(true(),x) -> double(1024_1(s(x)))
if(false(),x) -> s(0())
lt(0(),s(y)) -> true()
lt(x,0()) -> false()
lt(s(x),s(y)) -> lt(x,y)
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
10() -> double(s(double(s(s(0())))))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 1024() -> 1024_1(0())
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 10() -> double(s(double(s(s(0())))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: 1024^#() -> c_0(1024_1^#(0()))
, 2: 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
, 3: if^#(true(), x) -> c_2(double^#(1024_1(s(x))))
, 4: if^#(false(), x) -> c_3()
, 5: lt^#(0(), s(y)) -> c_4()
, 6: lt^#(x, 0()) -> c_5()
, 7: lt^#(s(x), s(y)) -> c_6(lt^#(x, y))
, 8: double^#(0()) -> c_7()
, 9: double^#(s(x)) -> c_8(double^#(x))
, 10: 10^#() -> c_9(double^#(s(double(s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ NA ]
|
`->{9} [ NA ]
|
`->{8} [ NA ]
->{7} [ YES(?,O(n^2)) ]
|
|->{5} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2} [ inherited ]
|
|->{3} [ inherited ]
| |
| |->{8} [ NA ]
| |
| `->{9} [ inherited ]
| |
| `->{8} [ NA ]
|
`->{4} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}->{8}: NA
---------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())}
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}->{2}->{3}->{9}: inherited
----------------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}->{9}->{8}: NA
--------------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())}
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}->{2}->{4}: MAYBE
-------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
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:
{ 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
, 1024^#() -> c_0(1024_1^#(0()))
, if^#(false(), x) -> c_3()
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
Proof Output:
The input cannot be shown compatible
* Path {7}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
10^#() = [0]
[0]
c_9(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: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_6(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {7}->{5}: YES(?,O(n^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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
10^#() = [0]
[0]
c_9(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: {lt^#(0(), s(y)) -> c_4()}
Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
lt^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_4() = [1]
[0]
c_6(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {7}->{6}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
10^#() = [0]
[0]
c_9(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: {lt^#(x, 0()) -> c_5()}
Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
lt^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_5() = [1]
[0]
c_6(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {10}: NA
-------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [2]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [3 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
10^#() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{9}: NA
------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [2]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [3 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [3]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [3 0] x1 + [0]
[3 3] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
10^#() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{9}->{8}: NA
-----------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [2]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [3 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [3]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
10^#() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: 1024^#() -> c_0(1024_1^#(0()))
, 2: 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
, 3: if^#(true(), x) -> c_2(double^#(1024_1(s(x))))
, 4: if^#(false(), x) -> c_3()
, 5: lt^#(0(), s(y)) -> c_4()
, 6: lt^#(x, 0()) -> c_5()
, 7: lt^#(s(x), s(y)) -> c_6(lt^#(x, y))
, 8: double^#(0()) -> c_7()
, 9: double^#(s(x)) -> c_8(double^#(x))
, 10: 10^#() -> c_9(double^#(s(double(s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(n^1)) ]
|
`->{9} [ NA ]
|
`->{8} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
`->{2} [ inherited ]
|
|->{3} [ inherited ]
| |
| |->{8} [ NA ]
| |
| `->{9} [ inherited ]
| |
| `->{8} [ NA ]
|
`->{4} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}->{8}: NA
---------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())}
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}->{2}->{3}->{9}: inherited
----------------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}->{9}->{8}: NA
--------------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())}
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}->{2}->{4}: MAYBE
-------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
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:
{ 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
, 1024^#() -> c_0(1024_1^#(0()))
, if^#(false(), x) -> c_3()
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
Proof Output:
The input cannot be shown compatible
* Path {7}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
10^#() = [0]
c_9(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: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
10^#() = [0]
c_9(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: {lt^#(0(), s(y)) -> c_4()}
Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [1]
c_6(x1) = [1] x1 + [7]
* Path {7}->{6}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
10^#() = [0]
c_9(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: {lt^#(x, 0()) -> c_5()}
Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_5() = [1]
c_6(x1) = [1] x1 + [7]
* Path {10}: YES(?,O(n^1))
------------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [1]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [3] x1 + [0]
s(x1) = [1] x1 + [1]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [3] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
10^#() = [0]
c_9(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {10^#() -> c_9(double^#(s(double(s(s(0()))))))}
Weak Rules:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
Proof Output:
The following argument positions are usable:
Uargs(double) = {}, Uargs(s) = {}, Uargs(double^#) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
double^#(x1) = [2] x1 + [0]
10^#() = [7]
c_9(x1) = [2] x1 + [3]
* Path {10}->{9}: NA
------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [1]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [3] x1 + [0]
s(x1) = [1] x1 + [1]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [3] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
10^#() = [0]
c_9(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{9}->{8}: NA
-----------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [1]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [3] x1 + [0]
s(x1) = [1] x1 + [1]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [3] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
10^#() = [0]
c_9(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ 1024() -> 1024_1(0())
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 10() -> double(s(double(s(s(0())))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: 1024^#() -> c_0(1024_1^#(0()))
, 2: 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
, 3: if^#(true(), x) -> c_2(double^#(1024_1(s(x))))
, 4: if^#(false(), x) -> c_3()
, 5: lt^#(0(), s(y)) -> c_4()
, 6: lt^#(x, 0()) -> c_5()
, 7: lt^#(s(x), s(y)) -> c_6(lt^#(x, y))
, 8: double^#(0()) -> c_7()
, 9: double^#(s(x)) -> c_8(double^#(x))
, 10: 10^#() -> c_9(double^#(s(double(s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ NA ]
|
`->{9} [ NA ]
|
`->{8} [ NA ]
->{7} [ YES(?,O(n^2)) ]
|
|->{5} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2} [ inherited ]
|
|->{3} [ inherited ]
| |
| |->{8} [ NA ]
| |
| `->{9} [ inherited ]
| |
| `->{8} [ NA ]
|
`->{4} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}->{8}: NA
---------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())}
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}->{2}->{3}->{9}: inherited
----------------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}->{9}->{8}: NA
--------------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())}
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}->{2}->{4}: MAYBE
-------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
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:
{ 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
, 1024^#() -> c_0(1024_1^#(0()))
, if^#(false(), x) -> c_3()
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
Proof Output:
The input cannot be shown compatible
* Path {7}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
10^#() = [0]
[0]
c_9(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: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_6(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {7}->{5}: YES(?,O(n^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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
10^#() = [0]
[0]
c_9(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: {lt^#(0(), s(y)) -> c_4()}
Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
lt^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_4() = [1]
[0]
c_6(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {7}->{6}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
10^#() = [0]
[0]
c_9(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: {lt^#(x, 0()) -> c_5()}
Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
lt^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_5() = [1]
[0]
c_6(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {10}: NA
-------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [2]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [3 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
10^#() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{9}: NA
------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [2]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [3 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [3]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [3 0] x1 + [0]
[3 3] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
10^#() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{9}->{8}: NA
-----------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
[0]
1024_1(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [2]
[0]
if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
10() = [0]
[0]
true() = [0]
[0]
double(x1) = [3 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [3]
[0 0] [0]
false() = [0]
[0]
1024^#() = [0]
[0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
1024_1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
10^#() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: 1024^#() -> c_0(1024_1^#(0()))
, 2: 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
, 3: if^#(true(), x) -> c_2(double^#(1024_1(s(x))))
, 4: if^#(false(), x) -> c_3()
, 5: lt^#(0(), s(y)) -> c_4()
, 6: lt^#(x, 0()) -> c_5()
, 7: lt^#(s(x), s(y)) -> c_6(lt^#(x, y))
, 8: double^#(0()) -> c_7()
, 9: double^#(s(x)) -> c_8(double^#(x))
, 10: 10^#() -> c_9(double^#(s(double(s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ YES(?,O(n^1)) ]
|
`->{9} [ NA ]
|
`->{8} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
`->{2} [ inherited ]
|
|->{3} [ inherited ]
| |
| |->{8} [ NA ]
| |
| `->{9} [ inherited ]
| |
| `->{8} [ NA ]
|
`->{4} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}: inherited
------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}: inherited
-----------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}->{8}: NA
---------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())}
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}->{2}->{3}->{9}: inherited
----------------------------------
This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
* Path {1}->{2}->{3}->{9}->{8}: NA
--------------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, 1024_1(x) -> if(lt(x, 10()), x)
, if(true(), x) -> double(1024_1(s(x)))
, if(false(), x) -> s(0())}
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}->{2}->{4}: MAYBE
-------------------------
The usable rules for this path are:
{ lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
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:
{ 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
, 1024^#() -> c_0(1024_1^#(0()))
, if^#(false(), x) -> c_3()
, lt(0(), s(y)) -> true()
, lt(x, 0()) -> false()
, lt(s(x), s(y)) -> lt(x, y)
, 10() -> double(s(double(s(s(0())))))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
Proof Output:
The input cannot be shown compatible
* Path {7}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
10^#() = [0]
c_9(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: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
10^#() = [0]
c_9(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: {lt^#(0(), s(y)) -> c_4()}
Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [1]
c_6(x1) = [1] x1 + [7]
* Path {7}->{6}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
10^#() = [0]
c_9(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: {lt^#(x, 0()) -> c_5()}
Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_5() = [1]
c_6(x1) = [1] x1 + [7]
* Path {10}: YES(?,O(n^1))
------------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [1]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [3] x1 + [0]
s(x1) = [1] x1 + [1]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [3] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
10^#() = [0]
c_9(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {10^#() -> c_9(double^#(s(double(s(s(0()))))))}
Weak Rules:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
Proof Output:
The following argument positions are usable:
Uargs(double) = {}, Uargs(s) = {}, Uargs(double^#) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
double^#(x1) = [2] x1 + [0]
10^#() = [7]
c_9(x1) = [2] x1 + [3]
* Path {10}->{9}: NA
------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [1]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [3] x1 + [0]
s(x1) = [1] x1 + [1]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [3] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
10^#() = [0]
c_9(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {10}->{9}->{8}: NA
-----------------------
The usable rules for this path are:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
1024() = [0]
1024_1(x1) = [0] x1 + [0]
0() = [1]
if(x1, x2) = [0] x1 + [0] x2 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
10() = [0]
true() = [0]
double(x1) = [3] x1 + [0]
s(x1) = [1] x1 + [1]
false() = [0]
1024^#() = [0]
c_0(x1) = [0] x1 + [0]
1024_1^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
if^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
double^#(x1) = [3] x1 + [0]
c_3() = [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
10^#() = [0]
c_9(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.