Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.39 |
|---|
stdout:
MAYBE
Problem:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.39 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | AG01 3.39 |
|---|
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, s(0())))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, s(0())))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: plus^#(0(), y) -> c_4()
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
, 7: plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
c_6(plus^#(minus(y, s(s(z))), minus(x, s(0()))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}
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]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {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(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}
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]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> 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:
0() = [2]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_0() = [1]
c_1(x1) = [1] x1 + [7]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))}
Weak Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [4]
quot^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_3(x1) = [1] x1 + [3]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y)) -> c_2()}
Weak Rules:
{ quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [1]
s(x1) = [1] x1 + [0]
quot^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
* Path {6,7}: YES(?,O(n^1))
-------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2] x1 + [0] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [2]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ plus^#(s(x), y) -> c_5(plus^#(x, y))
, plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
c_6(plus^#(minus(y, s(s(z))), minus(x, s(0()))))}
Weak Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [4] x1 + [2] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [2]
plus^#(x1, x2) = [2] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [7]
* Path {6,7}->{5}: YES(?,O(n^1))
------------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [2]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {plus^#(0(), y) -> c_4()}
Weak Rules:
{ plus^#(s(x), y) -> c_5(plus^#(x, y))
, plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
c_6(plus^#(minus(y, s(s(z))), minus(x, s(0()))))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2] x1 + [0] x2 + [4]
0() = [2]
s(x1) = [1] x1 + [4]
plus^#(x1, x2) = [2] x1 + [0] x2 + [0]
c_4() = [1]
c_5(x1) = [1] x1 + [7]
c_6(x1) = [0] x1 + [7]Tool RC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.39 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | AG01 3.39 |
|---|
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, s(0())))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, s(0())))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0(x)
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: plus^#(0(), y) -> c_4(y)
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
, 7: plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
c_6(plus^#(minus(y, s(s(z))), minus(x, s(0()))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,7} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
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]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {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(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
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]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_0(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_0) = {},
Uargs(c_1) = {1}
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_0(x1) = [0] x1 + [1]
c_1(x1) = [1] x1 + [5]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))}
Weak Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [4]
quot^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_3(x1) = [1] x1 + [3]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y)) -> c_2()}
Weak Rules:
{ quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [1]
s(x1) = [1] x1 + [0]
quot^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
* Path {6,7}: YES(?,O(n^1))
-------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2] x1 + [0] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [2]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ plus^#(s(x), y) -> c_5(plus^#(x, y))
, plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
c_6(plus^#(minus(y, s(s(z))), minus(x, s(0()))))}
Weak Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [4] x1 + [2] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [2]
plus^#(x1, x2) = [2] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [7]
* Path {6,7}->{5}: YES(?,O(n^1))
------------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2] x1 + [0] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [2]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {plus^#(0(), y) -> c_4(y)}
Weak Rules:
{ plus^#(s(x), y) -> c_5(plus^#(x, y))
, plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
c_6(plus^#(minus(y, s(s(z))), minus(x, s(0()))))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(plus^#) = {},
Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [2]
plus^#(x1, x2) = [2] x1 + [0] x2 + [2]
c_4(x1) = [0] x1 + [1]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [2]Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.39 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, 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 3.39 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, 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 3.39 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, 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 3.39 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, 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 3.39 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, 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 | 60.059155ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.39 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, plus(minus(x, s(0())), minus(y, s(s(z)))) ->
plus(minus(y, s(s(z))), minus(x, 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..