Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ from(X) -> cons(X, from(s(X)))
, length(nil()) -> 0()
, length(cons(X, Y)) -> s(length1(Y))
, length1(X) -> length(X)}
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: from^#(X) -> c_0(from^#(s(X)))
, 2: length^#(nil()) -> c_1()
, 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
, 4: length1^#(X) -> c_3(length^#(X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3,4} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {1}, Uargs(length^#) = {}, Uargs(c_2) = {},
Uargs(length1^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 0] [0]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
0() = [0]
[0]
[0]
length1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [3 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
length^#(x1) = [0 0 0] x1 + [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]
length1^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3,4}: YES(?,O(n^2))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
0() = [0]
[0]
[0]
length1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(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]
length^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
length1^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_3(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:
{ length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length1^#(X) -> c_3(length^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
length^#(x1) = [2 0 0] x1 + [0]
[6 0 0] [0]
[6 0 0] [0]
c_2(x1) = [1 0 0] x1 + [1]
[2 2 2] [0]
[2 0 0] [7]
length1^#(x1) = [2 2 2] x1 + [2]
[0 0 0] [2]
[0 0 2] [2]
c_3(x1) = [1 0 0] x1 + [1]
[0 0 0] [2]
[0 0 0] [2]
* Path {3,4}->{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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
0() = [0]
[0]
[0]
length1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(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]
length^#(x1) = [0 0 0] x1 + [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]
length1^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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: {length^#(nil()) -> c_1()}
Weak Rules:
{ length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length1^#(X) -> c_3(length^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 3 3] x2 + [0]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [0]
nil() = [2]
[0]
[2]
length^#(x1) = [2 0 2] x1 + [0]
[0 0 0] [0]
[4 2 0] [4]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 2 2] [0]
length1^#(x1) = [2 0 2] x1 + [0]
[0 2 1] [2]
[0 4 3] [2]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: from^#(X) -> c_0(from^#(s(X)))
, 2: length^#(nil()) -> c_1()
, 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
, 4: length1^#(X) -> c_3(length^#(X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3,4} [ NA ]
|
`->{2} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {1}, Uargs(length^#) = {}, Uargs(c_2) = {},
Uargs(length1^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
0() = [0]
[0]
length1(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
length1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [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: {from^#(X) -> c_0(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3,4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
0() = [0]
[0]
length1(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
length1^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We have not generated a proof for the resulting sub-problem.
* Path {3,4}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
0() = [0]
[0]
length1(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
length1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(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: from^#(X) -> c_0(from^#(s(X)))
, 2: length^#(nil()) -> c_1()
, 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
, 4: length1^#(X) -> c_3(length^#(X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3,4} [ NA ]
|
`->{2} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {1}, Uargs(length^#) = {}, Uargs(c_2) = {},
Uargs(length1^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
0() = [0]
length1(x1) = [0] x1 + [0]
from^#(x1) = [3] x1 + [0]
c_0(x1) = [1] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
length1^#(x1) = [0] x1 + [0]
c_3(x1) = [0] 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: {from^#(X) -> c_0(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3,4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
0() = [0]
length1(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
length^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
length1^#(x1) = [3] x1 + [0]
c_3(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {3,4}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
0() = [0]
length1(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
length^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
length1^#(x1) = [0] x1 + [0]
c_3(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:
{ from(X) -> cons(X, from(s(X)))
, length(nil()) -> 0()
, length(cons(X, Y)) -> s(length1(Y))
, length1(X) -> length(X)}
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: from^#(X) -> c_0(X, from^#(s(X)))
, 2: length^#(nil()) -> c_1()
, 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
, 4: length1^#(X) -> c_3(length^#(X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3,4} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {2}, Uargs(length^#) = {}, Uargs(c_2) = {},
Uargs(length1^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
s(x1) = [0 1 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
0() = [0]
[0]
[0]
length1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [1 3 3] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
length^#(x1) = [0 0 0] x1 + [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]
length1^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {from^#(X) -> c_0(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3,4}: YES(?,O(n^2))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 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]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
0() = [0]
[0]
[0]
length1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
length^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
length1^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_3(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:
{ length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length1^#(X) -> c_3(length^#(X))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
length^#(x1) = [2 0 0] x1 + [0]
[6 0 0] [0]
[6 0 0] [0]
c_2(x1) = [1 0 0] x1 + [1]
[2 2 2] [0]
[2 0 0] [7]
length1^#(x1) = [2 2 2] x1 + [2]
[0 0 0] [2]
[0 0 2] [2]
c_3(x1) = [1 0 0] x1 + [1]
[0 0 0] [2]
[0 0 0] [2]
* Path {3,4}->{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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
length(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
0() = [0]
[0]
[0]
length1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(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]
length^#(x1) = [0 0 0] x1 + [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]
length1^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(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: {length^#(nil()) -> c_1()}
Weak Rules:
{ length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length1^#(X) -> c_3(length^#(X))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 3 3] x2 + [0]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [0]
nil() = [2]
[0]
[2]
length^#(x1) = [2 0 2] x1 + [0]
[0 0 0] [0]
[4 2 0] [4]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 2 2] [0]
length1^#(x1) = [2 0 2] x1 + [0]
[0 2 1] [2]
[0 4 3] [2]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: from^#(X) -> c_0(X, from^#(s(X)))
, 2: length^#(nil()) -> c_1()
, 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
, 4: length1^#(X) -> c_3(length^#(X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3,4} [ NA ]
|
`->{2} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {2}, Uargs(length^#) = {}, Uargs(c_2) = {},
Uargs(length1^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
0() = [0]
[0]
length1(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_0(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
length1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [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: {from^#(X) -> c_0(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3,4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
0() = [0]
[0]
length1(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
length1^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We have not generated a proof for the resulting sub-problem.
* Path {3,4}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
length(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
0() = [0]
[0]
length1(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
length^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
length1^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(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: from^#(X) -> c_0(X, from^#(s(X)))
, 2: length^#(nil()) -> c_1()
, 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
, 4: length1^#(X) -> c_3(length^#(X))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3,4} [ NA ]
|
`->{2} [ NA ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {2}, Uargs(length^#) = {}, Uargs(c_2) = {},
Uargs(length1^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
0() = [0]
length1(x1) = [0] x1 + [0]
from^#(x1) = [3] x1 + [0]
c_0(x1, x2) = [2] x1 + [1] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
length1^#(x1) = [0] x1 + [0]
c_3(x1) = [0] 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: {from^#(X) -> c_0(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
* Path {3,4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
0() = [0]
length1(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
length1^#(x1) = [3] x1 + [0]
c_3(x1) = [1] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {3,4}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
Uargs(length1^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
from(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
length(x1) = [0] x1 + [0]
nil() = [0]
0() = [0]
length1(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_0(x1, x2) = [0] x1 + [0] x2 + [0]
length^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
length1^#(x1) = [0] x1 + [0]
c_3(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.