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