Tool CaT
stdout:
MAYBE
Problem:
f(x,empty()) -> x
f(empty(),cons(a,k)) -> f(cons(a,k),k)
f(cons(a,k),y) -> f(y,k)
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:
{ f(x, empty()) -> x
, f(empty(), cons(a, k)) -> f(cons(a, k), k)
, f(cons(a, k), y) -> f(y, k)}
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: f^#(x, empty()) -> c_0()
, 2: f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, 3: f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
empty() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [1 3 3] x2 + [0]
[0 1 3] [0 1 3] [0]
[0 0 1] [0 0 1] [0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[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:
{ f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
empty() = [0]
[0]
[0]
cons(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]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(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 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: f^#(x, empty()) -> c_0()
, 2: f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, 3: f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
empty() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[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:
{ f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
empty() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(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: f^#(x, empty()) -> c_0()
, 2: f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, 3: f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [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:
{ f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [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:
{ f(x, empty()) -> x
, f(empty(), cons(a, k)) -> f(cons(a, k), k)
, f(cons(a, k), y) -> f(y, k)}
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: f^#(x, empty()) -> c_0(x)
, 2: f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, 3: f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
empty() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [1 3 3] x2 + [0]
[0 1 3] [0 1 3] [0]
[0 0 1] [0 0 1] [0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[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:
{ f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(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]
empty() = [0]
[0]
[0]
cons(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]
f^#(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_0(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[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 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: f^#(x, empty()) -> c_0(x)
, 2: f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, 3: f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
empty() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[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:
{ f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
empty() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[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: f^#(x, empty()) -> c_0(x)
, 2: f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, 3: f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [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:
{ f^#(empty(), cons(a, k)) -> c_1(f^#(cons(a, k), k))
, f^#(cons(a, k), y) -> c_2(f^#(y, k))}
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(f) = {}, Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [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.