Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.8a |
|---|
stdout:
MAYBE
Problem:
pred(s(x)) -> x
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.8a |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.8a |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ pred(s(x)) -> x
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(quot(x, s(s(0()))))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: pred^#(s(x)) -> c_0()
, 2: minus^#(x, 0()) -> c_1()
, 3: minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)))
, 4: quot^#(0(), s(y)) -> c_3()
, 5: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, 6: log^#(s(0())) -> c_5()
, 7: log^#(s(s(x))) -> c_6(log^#(s(quot(x, s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{6} [ NA ]
->{5} [ MAYBE ]
|
`->{4} [ NA ]
->{3} [ NA ]
|
`->{1} [ NA ]
->{2} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {2}: 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(pred) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(minus^#) = {}, Uargs(c_2) = {}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus(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]
quot(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]
log(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pred^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
minus^#(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_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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: {minus^#(x, 0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(minus^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
minus^#(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_1() = [0]
[1]
[1]
* Path {3}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(minus^#) = {}, Uargs(c_2) = {1}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
minus(x1, x2) = [3 0 0] x1 + [1 0 0] x2 + [2]
[0 2 0] [0 0 0] [0]
[0 0 2] [0 1 0] [0]
0() = [0]
[0]
[0]
quot(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]
log(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pred^#(x1) = [1 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0() = [0]
[0]
[0]
minus^#(x1, x2) = [3 3 3] x1 + [2 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(minus^#) = {}, Uargs(c_2) = {1}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 1 0] [0]
s(x1) = [1 0 3] x1 + [3]
[0 1 1] [0]
[0 0 0] [2]
minus(x1, x2) = [2 0 0] x1 + [0 2 2] x2 + [0]
[0 2 0] [3 0 0] [0]
[0 2 2] [3 0 0] [0]
0() = [0]
[1]
[0]
quot(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]
log(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pred^#(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
minus^#(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_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {5}: MAYBE
---------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {5}->{4}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(minus^#) = {}, Uargs(c_2) = {}, Uargs(quot^#) = {1},
Uargs(c_4) = {1}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 1 0] [0]
s(x1) = [1 0 3] x1 + [3]
[0 1 1] [0]
[0 0 0] [2]
minus(x1, x2) = [2 0 0] x1 + [0 2 2] x2 + [0]
[0 2 0] [3 0 0] [0]
[0 2 2] [3 0 0] [0]
0() = [0]
[1]
[0]
quot(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]
log(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pred^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
minus^#(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_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [3 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(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
log^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{6}.
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
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: pred^#(s(x)) -> c_0()
, 2: minus^#(x, 0()) -> c_1()
, 3: minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)))
, 4: quot^#(0(), s(y)) -> c_3()
, 5: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, 6: log^#(s(0())) -> c_5()
, 7: log^#(s(s(x))) -> c_6(log^#(s(quot(x, s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{6} [ NA ]
->{5} [ MAYBE ]
|
`->{4} [ NA ]
->{3} [ NA ]
|
`->{1} [ NA ]
->{2} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {2}: 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(pred) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(minus^#) = {}, Uargs(c_2) = {}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
pred^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [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: {minus^#(x, 0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(minus^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
minus^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_1() = [0]
[1]
* Path {3}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(minus^#) = {}, Uargs(c_2) = {1}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0] x1 + [3]
[0 1] [3]
s(x1) = [1 3] x1 + [0]
[0 1] [2]
minus(x1, x2) = [1 1] x1 + [0 3] x2 + [2]
[0 3] [1 2] [0]
0() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
pred^#(x1) = [2 0] x1 + [0]
[3 3] [0]
c_0() = [0]
[0]
minus^#(x1, x2) = [3 3] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(minus^#) = {}, Uargs(c_2) = {1}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0] x1 + [0]
[1 0] [0]
s(x1) = [1 1] x1 + [1]
[0 0] [0]
minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[2 1] [1 0] [0]
0() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
pred^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5}: MAYBE
---------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {5}->{4}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(minus^#) = {}, Uargs(c_2) = {}, Uargs(quot^#) = {1},
Uargs(c_4) = {1}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0] x1 + [0]
[1 0] [0]
s(x1) = [1 1] x1 + [1]
[0 0] [0]
minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[2 1] [1 0] [0]
0() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
pred^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{6}.
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
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: pred^#(s(x)) -> c_0()
, 2: minus^#(x, 0()) -> c_1()
, 3: minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)))
, 4: quot^#(0(), s(y)) -> c_3()
, 5: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, 6: log^#(s(0())) -> c_5()
, 7: log^#(s(s(x))) -> c_6(log^#(s(quot(x, s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{6} [ NA ]
->{5} [ MAYBE ]
|
`->{4} [ NA ]
->{3} [ NA ]
|
`->{1} [ NA ]
->{2} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {2}: 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(pred) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(minus^#) = {}, Uargs(c_2) = {}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1) = [0] x1 + [0]
pred^#(x1) = [0] x1 + [0]
c_0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
log^#(x1) = [0] x1 + [0]
c_5() = [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: {minus^#(x, 0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(minus^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
minus^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_1() = [1]
* Path {3}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(minus^#) = {}, Uargs(c_2) = {1}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1] x1 + [3]
s(x1) = [1] x1 + [2]
minus(x1, x2) = [3] x1 + [2] x2 + [0]
0() = [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1) = [0] x1 + [0]
pred^#(x1) = [1] x1 + [0]
c_0() = [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
log^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(minus^#) = {}, Uargs(c_2) = {1}, Uargs(quot^#) = {},
Uargs(c_4) = {}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1] x1 + [3]
s(x1) = [1] x1 + [2]
minus(x1, x2) = [3] x1 + [2] x2 + [0]
0() = [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1) = [0] x1 + [0]
pred^#(x1) = [3] x1 + [0]
c_0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
log^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5}: MAYBE
---------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {5}->{4}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(minus^#) = {}, Uargs(c_2) = {}, Uargs(quot^#) = {1},
Uargs(c_4) = {1}, Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1] x1 + [3]
s(x1) = [1] x1 + [2]
minus(x1, x2) = [3] x1 + [2] x2 + [0]
0() = [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1) = [0] x1 + [0]
pred^#(x1) = [0] x1 + [0]
c_0() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
log^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{6}.
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
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
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.8a |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.8a |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ pred(s(x)) -> x
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(quot(x, s(s(0()))))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: pred^#(s(x)) -> c_0(x)
, 2: minus^#(x, 0()) -> c_1(x)
, 3: minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)))
, 4: quot^#(0(), s(y)) -> c_3()
, 5: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, 6: log^#(s(0())) -> c_5()
, 7: log^#(s(s(x))) -> c_6(log^#(s(quot(x, s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{6} [ NA ]
->{5} [ MAYBE ]
|
`->{4} [ NA ]
->{3} [ NA ]
|
`->{1} [ NA ]
->{2} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {2}: 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(pred) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus(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]
quot(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]
log(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pred^#(x1) = [0 0 0] x1 + [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]
minus^#(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_1(x1) = [1 1 1] x1 + [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]
quot^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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: {minus^#(x, 0()) -> c_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(minus^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[2]
minus^#(x1, x2) = [7 7 7] x1 + [2 0 2] x2 + [7]
[7 7 7] [2 0 2] [7]
[7 7 7] [2 0 2] [7]
c_1(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {3}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {1}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
minus(x1, x2) = [3 0 0] x1 + [1 0 0] x2 + [2]
[0 2 0] [0 0 0] [0]
[0 0 2] [0 1 0] [0]
0() = [0]
[0]
[0]
quot(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]
log(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pred^#(x1) = [1 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(x1, x2) = [3 3 3] x1 + [2 0 0] x2 + [0]
[0 0 0] [0 0 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]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {1}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
s(x1) = [1 1 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
minus(x1, x2) = [2 0 0] x1 + [3 0 0] x2 + [2]
[0 2 0] [1 0 0] [0]
[0 0 2] [3 0 0] [0]
0() = [0]
[0]
[0]
quot(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]
log(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pred^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
log^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {5}: MAYBE
---------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {5}->{4}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(quot^#) = {1}, Uargs(c_4) = {1},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 1 0] [0]
s(x1) = [1 0 3] x1 + [3]
[0 1 1] [0]
[0 0 0] [2]
minus(x1, x2) = [2 0 0] x1 + [0 2 2] x2 + [0]
[0 2 0] [3 0 0] [0]
[0 2 2] [3 0 0] [0]
0() = [0]
[1]
[0]
quot(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]
log(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
pred^#(x1) = [0 0 0] x1 + [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]
minus^#(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_1(x1) = [0 0 0] x1 + [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]
quot^#(x1, x2) = [3 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(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
log^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{6}.
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
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: pred^#(s(x)) -> c_0(x)
, 2: minus^#(x, 0()) -> c_1(x)
, 3: minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)))
, 4: quot^#(0(), s(y)) -> c_3()
, 5: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, 6: log^#(s(0())) -> c_5()
, 7: log^#(s(s(x))) -> c_6(log^#(s(quot(x, s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{6} [ NA ]
->{5} [ MAYBE ]
|
`->{4} [ NA ]
->{3} [ NA ]
|
`->{1} [ NA ]
->{2} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {2}: 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(pred) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
pred^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 1] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [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: {minus^#(x, 0()) -> c_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(minus^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
minus^#(x1, x2) = [7 7] x1 + [2 2] x2 + [7]
[7 7] [2 2] [3]
c_1(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {3}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {1}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0] x1 + [3]
[0 1] [3]
s(x1) = [1 3] x1 + [0]
[0 1] [2]
minus(x1, x2) = [1 1] x1 + [0 3] x2 + [2]
[0 3] [1 2] [0]
0() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
pred^#(x1) = [2 0] x1 + [0]
[3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {1}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0] x1 + [1]
[0 1] [0]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
minus(x1, x2) = [3 3] x1 + [2 0] x2 + [2]
[3 3] [0 0] [0]
0() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
pred^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5}: MAYBE
---------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {5}->{4}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(quot^#) = {1}, Uargs(c_4) = {1},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1 0] x1 + [0]
[1 0] [0]
s(x1) = [1 1] x1 + [1]
[0 0] [0]
minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[2 1] [1 0] [0]
0() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
pred^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{6}.
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
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: pred^#(s(x)) -> c_0(x)
, 2: minus^#(x, 0()) -> c_1(x)
, 3: minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)))
, 4: quot^#(0(), s(y)) -> c_3()
, 5: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, 6: log^#(s(0())) -> c_5()
, 7: log^#(s(s(x))) -> c_6(log^#(s(quot(x, s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{6} [ NA ]
->{5} [ MAYBE ]
|
`->{4} [ NA ]
->{3} [ NA ]
|
`->{1} [ NA ]
->{2} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {2}: 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(pred) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1) = [0] x1 + [0]
pred^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
log^#(x1) = [0] x1 + [0]
c_5() = [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: {minus^#(x, 0()) -> c_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(minus^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [5]
minus^#(x1, x2) = [7] x1 + [3] x2 + [0]
c_1(x1) = [1] x1 + [0]
* Path {3}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {1}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1] x1 + [3]
s(x1) = [1] x1 + [2]
minus(x1, x2) = [3] x1 + [2] x2 + [0]
0() = [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1) = [0] x1 + [0]
pred^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
log^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {1},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {1}, Uargs(quot^#) = {}, Uargs(c_4) = {},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [2]
minus(x1, x2) = [3] x1 + [2] x2 + [0]
0() = [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1) = [0] x1 + [0]
pred^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
log^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {5}: MAYBE
---------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {5}->{4}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(pred) = {1}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(quot) = {}, Uargs(log) = {}, Uargs(pred^#) = {},
Uargs(c_0) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(c_2) = {}, Uargs(quot^#) = {1}, Uargs(c_4) = {1},
Uargs(log^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
pred(x1) = [1] x1 + [3]
s(x1) = [1] x1 + [2]
minus(x1, x2) = [3] x1 + [2] x2 + [0]
0() = [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
log(x1) = [0] x1 + [0]
pred^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
log^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{6}.
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, pred(s(x)) -> x}
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 pair1rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.8a |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ pred(s(x)) -> x
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(quot(x, s(s(0()))))))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.8a |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ pred(s(x)) -> x
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(quot(x, s(s(0()))))))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.8a |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ pred(s(x)) -> x
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(quot(x, s(s(0()))))))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.8a |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ pred(s(x)) -> x
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(quot(x, s(s(0()))))))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.8a |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ pred(s(x)) -> x
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(quot(x, s(s(0()))))))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
| Execution Time | 60.853558ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.8a |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ pred(s(x)) -> x
, minus(x, 0()) -> x
, minus(x, s(y)) -> pred(minus(x, y))
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(quot(x, s(s(0()))))))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..