Tool CaT
stdout:
MAYBE
Problem:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(0()),y) -> y
times(s(x),y) -> plus(y,times(x,y))
div(0(),y) -> 0()
div(x,y) -> quot(x,y,y)
quot(0(),s(y),z) -> 0()
quot(s(x),s(y),z) -> quot(x,y,z)
quot(x,0(),s(z)) -> s(div(x,s(z)))
div(div(x,y),z) -> div(x,times(y,z))
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:
{ plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, div(0(), y) -> 0()
, div(x, y) -> quot(x, y, y)
, quot(0(), s(y), z) -> 0()
, quot(s(x), s(y), z) -> quot(x, y, z)
, quot(x, 0(), s(z)) -> s(div(x, s(z)))
, div(div(x, y), z) -> div(x, times(y, z))}
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: plus^#(x, 0()) -> c_0()
, 2: plus^#(0(), y) -> c_1()
, 3: plus^#(s(x), y) -> c_2(plus^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(0()), y) -> c_4()
, 6: times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
, 7: div^#(0(), y) -> c_6()
, 8: div^#(x, y) -> c_7(quot^#(x, y, y))
, 9: quot^#(0(), s(y), z) -> c_8()
, 10: quot^#(s(x), s(y), z) -> c_9(quot^#(x, y, z))
, 11: quot^#(x, 0(), s(z)) -> c_10(div^#(x, s(z)))
, 12: div^#(div(x, y), z) -> c_11(div^#(x, times(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,12,11,10} [ inherited ]
|
|->{7} [ NA ]
|
`->{9} [ NA ]
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
|->{2} [ NA ]
|
`->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {4}: 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
div(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]
quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div^#(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11(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: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
times^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
[2 2 0] [0 0 0] [3]
[2 2 2] [0 0 0] [3]
c_3() = [0]
[1]
[1]
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
div(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]
quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div^#(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11(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: {times^#(s(0()), y) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [0 3 2] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [3]
[0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [7]
c_4() = [0]
[1]
[1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We 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:
{ times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
, plus^#(x, 0()) -> c_0()
, times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: NA
-----------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{3}: inherited
------------------------
This path is subsumed by the proof of path {6}->{3}->{1}.
* Path {6}->{3}->{1}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,12,11,10}: inherited
----------------------------
This path is subsumed by the proof of path {8,12,11,10}->{7}.
* Path {8,12,11,10}->{7}: NA
--------------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,12,11,10}->{9}: NA
--------------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, 0()) -> c_0()
, 2: plus^#(0(), y) -> c_1()
, 3: plus^#(s(x), y) -> c_2(plus^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(0()), y) -> c_4()
, 6: times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
, 7: div^#(0(), y) -> c_6()
, 8: div^#(x, y) -> c_7(quot^#(x, y, y))
, 9: quot^#(0(), s(y), z) -> c_8()
, 10: quot^#(s(x), s(y), z) -> c_9(quot^#(x, y, z))
, 11: quot^#(x, 0(), s(z)) -> c_10(div^#(x, s(z)))
, 12: div^#(div(x, y), z) -> c_11(div^#(x, times(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,12,11,10} [ inherited ]
|
|->{7} [ NA ]
|
`->{9} [ NA ]
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
|->{2} [ NA ]
|
`->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {4}: 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
times^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
[2 2] [0 0] [7]
c_3() = [0]
[1]
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(s(0()), y) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [0 1] x1 + [2]
[0 0] [2]
times^#(x1, x2) = [2 2] x1 + [0 0] x2 + [3]
[2 2] [0 0] [3]
c_4() = [0]
[1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We 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:
{ times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
, plus^#(x, 0()) -> c_0()
, times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: NA
-----------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{3}: inherited
------------------------
This path is subsumed by the proof of path {6}->{3}->{1}.
* Path {6}->{3}->{1}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,12,11,10}: inherited
----------------------------
This path is subsumed by the proof of path {8,12,11,10}->{7}.
* Path {8,12,11,10}->{7}: NA
--------------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,12,11,10}->{9}: NA
--------------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(x, 0()) -> c_0()
, 2: plus^#(0(), y) -> c_1()
, 3: plus^#(s(x), y) -> c_2(plus^#(x, y))
, 4: times^#(0(), y) -> c_3()
, 5: times^#(s(0()), y) -> c_4()
, 6: times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
, 7: div^#(0(), y) -> c_6()
, 8: div^#(x, y) -> c_7(quot^#(x, y, y))
, 9: quot^#(0(), s(y), z) -> c_8()
, 10: quot^#(s(x), s(y), z) -> c_9(quot^#(x, y, z))
, 11: quot^#(x, 0(), s(z)) -> c_10(div^#(x, s(z)))
, 12: div^#(div(x, y), z) -> c_11(div^#(x, times(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,12,11,10} [ inherited ]
|
|->{7} [ NA ]
|
`->{9} [ NA ]
->{6} [ inherited ]
|
|->{1} [ MAYBE ]
|
|->{2} [ NA ]
|
`->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ NA ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {4}: 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(0(), y) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [1] x1 + [0] x2 + [7]
c_3() = [1]
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(s(0()), y) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
times^#(x1, x2) = [2] x1 + [0] x2 + [7]
c_4() = [0]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{3}->{1}.
* Path {6}->{1}: MAYBE
--------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We 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:
{ times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
, plus^#(x, 0()) -> c_0()
, times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{2}: NA
-----------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{3}: inherited
------------------------
This path is subsumed by the proof of path {6}->{3}->{1}.
* Path {6}->{3}->{1}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{3}->{2}: NA
----------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,12,11,10}: inherited
----------------------------
This path is subsumed by the proof of path {8,12,11,10}->{7}.
* Path {8,12,11,10}->{7}: NA
--------------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,12,11,10}->{9}: NA
--------------------------
The usable rules for this path are:
{ times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
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:
TIMEOUT
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: TIMEOUT
Input Problem: runtime-complexity with respect to
Rules:
{ plus(x, 0()) -> x
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, times(0(), y) -> 0()
, times(s(0()), y) -> y
, times(s(x), y) -> plus(y, times(x, y))
, div(0(), y) -> 0()
, div(x, y) -> quot(x, y, y)
, quot(0(), s(y), z) -> 0()
, quot(s(x), s(y), z) -> quot(x, y, z)
, quot(x, 0(), s(z)) -> s(div(x, s(z)))
, div(div(x, y), z) -> div(x, times(y, z))}
Proof Output:
Computation stopped due to timeout after 60.0 seconds