Tool CaT
stdout:
MAYBE
Problem:
quot(0(),s(y),s(z)) -> 0()
quot(s(x),s(y),z) -> quot(x,y,z)
quot(x,0(),s(z)) -> s(quot(x,s(z),s(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:
{ quot(0(), s(y), s(z)) -> 0()
, quot(s(x), s(y), z) -> quot(x, y, z)
, quot(x, 0(), s(z)) -> s(quot(x, s(z), s(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: quot^#(0(), s(y), s(z)) -> c_0()
, 2: quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, 3: quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ MAYBE ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2,3}: MAYBE
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
0() = [0]
[0]
[0]
s(x1) = [1 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]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,3}->{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(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
0() = [0]
[0]
[0]
s(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_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y), s(z)) -> c_0()}
Weak Rules:
{ quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(quot^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 0 4] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2, x3) = [2 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[1 2 4] [0 0 0] [2 0 0] [0]
[0 0 0] [0 0 4] [0 0 0] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [7]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: quot^#(0(), s(y), s(z)) -> c_0()
, 2: quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, 3: quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ MAYBE ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2,3}: MAYBE
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 1] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,3}->{1}: NA
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
s(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_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
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: quot^#(0(), s(y), s(z)) -> c_0()
, 2: quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, 3: quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ MAYBE ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2,3}: MAYBE
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
quot^#(x1, x2, x3) = [3] x1 + [0] x2 + [3] x3 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,3}->{1}: NA
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ quot(0(), s(y), s(z)) -> 0()
, quot(s(x), s(y), z) -> quot(x, y, z)
, quot(x, 0(), s(z)) -> s(quot(x, s(z), s(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: quot^#(0(), s(y), s(z)) -> c_0()
, 2: quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, 3: quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ MAYBE ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2,3}: MAYBE
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
0() = [0]
[0]
[0]
s(x1) = [1 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]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,3}->{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(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
0() = [0]
[0]
[0]
s(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_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y), s(z)) -> c_0()}
Weak Rules:
{ quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(quot^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 0 4] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2, x3) = [2 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[1 2 4] [0 0 0] [2 0 0] [0]
[0 0 0] [0 0 4] [0 0 0] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [7]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: quot^#(0(), s(y), s(z)) -> c_0()
, 2: quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, 3: quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ MAYBE ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2,3}: MAYBE
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 1] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,3}->{1}: NA
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
s(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_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
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: quot^#(0(), s(y), s(z)) -> c_0()
, 2: quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, 3: quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ MAYBE ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2,3}: MAYBE
-----------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
quot^#(x1, x2, x3) = [3] x1 + [0] x2 + [3] x3 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ quot^#(s(x), s(y), z) -> c_1(quot^#(x, y, z))
, quot^#(x, 0(), s(z)) -> c_2(quot^#(x, s(z), s(z)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {2,3}->{1}: NA
-------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(quot) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
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.