Tool CaT
stdout:
MAYBE
Problem:
minus(x,x) -> 0()
minus(s(x),s(y)) -> minus(x,y)
minus(0(),x) -> 0()
minus(x,0()) -> x
div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
div(0(),s(y)) -> 0()
f(x,0(),b) -> x
f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)
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:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 0()
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)}
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: minus^#(x, x) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: minus^#(0(), x) -> c_2()
, 4: minus^#(x, 0()) -> c_3()
, 5: div^#(s(x), s(y)) -> c_4(div^#(minus(x, y), s(y)))
, 6: div^#(0(), s(y)) -> c_5()
, 7: f^#(x, 0(), b) -> c_6()
, 8: f^#(x, s(y), b) ->
c_7(div^#(f(x, minus(s(y), s(0())), b), b))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
|->{5} [ inherited ]
| |
| `->{6} [ NA ]
|
`->{6} [ NA ]
->{7} [ YES(?,O(1)) ]
->{2} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
minus^#(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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(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: {minus^#(x, x) -> c_0()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 4] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {2}->{3}: 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(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: {minus^#(0(), x) -> c_2()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [7]
c_2() = [1]
[0]
* Path {2}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(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: {minus^#(x, 0()) -> c_3()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, 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]
minus^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_1(x1) = [1 0] x1 + [6]
[0 0] [7]
c_3() = [1]
[0]
* Path {7}: 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(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: {f^#(x, 0(), b) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
f^#(x1, x2, x3) = [0 0] x1 + [2 0] x2 + [0 0] x3 + [7]
[0 0] [2 2] [0 0] [7]
c_6() = [0]
[1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{5}->{6}.
* Path {8}->{5}: inherited
------------------------
This path is subsumed by the proof of path {8}->{5}->{6}.
* Path {8}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 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 {8}->{6}: NA
-----------------
The usable rules for this path are:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, x) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: minus^#(0(), x) -> c_2()
, 4: minus^#(x, 0()) -> c_3()
, 5: div^#(s(x), s(y)) -> c_4(div^#(minus(x, y), s(y)))
, 6: div^#(0(), s(y)) -> c_5()
, 7: f^#(x, 0(), b) -> c_6()
, 8: f^#(x, s(y), b) ->
c_7(div^#(f(x, minus(s(y), s(0())), b), b))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
|->{5} [ inherited ]
| |
| `->{6} [ NA ]
|
`->{6} [ MAYBE ]
->{7} [ YES(?,O(1)) ]
->{2} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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: {minus^#(x, x) -> c_0()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [2]
c_0() = [1]
c_1(x1) = [1] x1 + [5]
* Path {2}->{3}: 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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: {minus^#(0(), x) -> c_2()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1(x1) = [1] x1 + [2]
c_2() = [1]
* Path {2}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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: {minus^#(x, 0()) -> c_3()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_1(x1) = [1] x1 + [7]
c_3() = [1]
* Path {7}: 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {}, Uargs(div^#) = {},
Uargs(c_4) = {}, Uargs(f^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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: {f^#(x, 0(), b) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
f^#(x1, x2, x3) = [0] x1 + [1] x2 + [0] x3 + [7]
c_6() = [1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{5}->{6}.
* Path {8}->{5}: inherited
------------------------
This path is subsumed by the proof of path {8}->{5}->{6}.
* Path {8}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 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 {8}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 0()}
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:
{ f^#(x, s(y), b) -> c_7(div^#(f(x, minus(s(y), s(0())), b), b))
, div^#(0(), s(y)) -> c_5()
, minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 0()}
Proof Output:
The input cannot be shown compatible
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:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 0()
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)}
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: minus^#(x, x) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: minus^#(0(), x) -> c_2()
, 4: minus^#(x, 0()) -> c_3(x)
, 5: div^#(s(x), s(y)) -> c_4(div^#(minus(x, y), s(y)))
, 6: div^#(0(), s(y)) -> c_5()
, 7: f^#(x, 0(), b) -> c_6(x)
, 8: f^#(x, s(y), b) ->
c_7(div^#(f(x, minus(s(y), s(0())), b), b))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
|->{5} [ inherited ]
| |
| `->{6} [ NA ]
|
`->{6} [ NA ]
->{7} [ YES(?,O(1)) ]
->{2} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
minus^#(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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(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: {minus^#(x, x) -> c_0()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 4] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {2}->{3}: 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(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: {minus^#(0(), x) -> c_2()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [7]
c_2() = [1]
[0]
* Path {2}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
c_3(x1) = [1 1] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(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: {minus^#(x, 0()) -> c_3(x)}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
[4 1] [3 2] [0]
c_1(x1) = [1 0] x1 + [0]
[2 0] [0]
c_3(x1) = [0 0] x1 + [1]
[0 1] [0]
* Path {7}: 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
minus^#(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]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
f^#(x1, x2, x3) = [3 3] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6(x1) = [1 1] x1 + [0]
[0 0] [0]
c_7(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: {f^#(x, 0(), b) -> c_6(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
f^#(x1, x2, x3) = [7 7] x1 + [2 2] x2 + [0 0] x3 + [7]
[7 7] [2 2] [0 0] [3]
c_6(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{5}->{6}.
* Path {8}->{5}: inherited
------------------------
This path is subsumed by the proof of path {8}->{5}->{6}.
* Path {8}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 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 {8}->{6}: NA
-----------------
The usable rules for this path are:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, x) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: minus^#(0(), x) -> c_2()
, 4: minus^#(x, 0()) -> c_3(x)
, 5: div^#(s(x), s(y)) -> c_4(div^#(minus(x, y), s(y)))
, 6: div^#(0(), s(y)) -> c_5()
, 7: f^#(x, 0(), b) -> c_6(x)
, 8: f^#(x, s(y), b) ->
c_7(div^#(f(x, minus(s(y), s(0())), b), b))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8} [ inherited ]
|
|->{5} [ inherited ]
| |
| `->{6} [ NA ]
|
`->{6} [ MAYBE ]
->{7} [ YES(?,O(1)) ]
->{2} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6(x1) = [0] x1 + [0]
c_7(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6(x1) = [0] x1 + [0]
c_7(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: {minus^#(x, x) -> c_0()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [2]
c_0() = [1]
c_1(x1) = [1] x1 + [5]
* Path {2}->{3}: 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6(x1) = [0] x1 + [0]
c_7(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: {minus^#(0(), x) -> c_2()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1(x1) = [1] x1 + [2]
c_2() = [1]
* Path {2}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {1}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6(x1) = [0] x1 + [0]
c_7(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: {minus^#(x, 0()) -> c_3(x)}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [2]
c_1(x1) = [1] x1 + [5]
c_3(x1) = [0] x1 + [1]
* Path {7}: 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(minus) = {}, Uargs(s) = {}, Uargs(div) = {}, Uargs(f) = {},
Uargs(minus^#) = {}, Uargs(c_1) = {}, Uargs(c_3) = {},
Uargs(div^#) = {}, Uargs(c_4) = {}, Uargs(f^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
f^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_6(x1) = [1] x1 + [0]
c_7(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: {f^#(x, 0(), b) -> c_6(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(f^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [5]
f^#(x1, x2, x3) = [7] x1 + [3] x2 + [0] x3 + [0]
c_6(x1) = [1] x1 + [0]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{5}->{6}.
* Path {8}->{5}: inherited
------------------------
This path is subsumed by the proof of path {8}->{5}->{6}.
* Path {8}->{5}->{6}: NA
----------------------
The usable rules for this path are:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 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 {8}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 0()}
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:
{ f^#(x, s(y), b) -> c_7(div^#(f(x, minus(s(y), s(0())), b), b))
, div^#(0(), s(y)) -> c_5()
, minus(x, x) -> 0()
, minus(s(x), s(y)) -> minus(x, y)
, minus(0(), x) -> 0()
, minus(x, 0()) -> x
, f(x, 0(), b) -> x
, f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b)
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(0(), s(y)) -> 0()}
Proof Output:
The input cannot be shown compatible
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.