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:
{ first(0(), X) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
, from(X) -> cons(X, from(s(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: first^#(0(), X) -> c_0()
, 2: first^#(s(X), cons(Y, Z)) -> c_1(first^#(X, Z))
, 3: from^#(X) -> c_2(from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ MAYBE ]
->{2} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {1},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(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]
nil() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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]
from^#(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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_1(first^#(X, Z))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 1] [2]
first^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
[2 0 2] [2 0 0] [0]
[0 2 2] [1 0 0] [0]
c_1(x1) = [1 0 0] x1 + [5]
[2 0 2] [3]
[0 0 0] [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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {1},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(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]
nil() = [0]
[0]
[0]
s(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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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]
from^#(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]
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: {first^#(0(), X) -> c_0()}
Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_1(first^#(X, Z))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [0]
[0 0 0] [0 0 4] [0]
[0 0 0] [0 0 0] [0]
first^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
[2 0 0] [2 0 0] [4]
[0 0 0] [0 2 0] [2]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {},
Uargs(from^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(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]
nil() = [0]
[0]
[0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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) = [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_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: {from^#(X) -> c_2(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: first^#(0(), X) -> c_0()
, 2: first^#(s(X), cons(Y, Z)) -> c_1(first^#(X, Z))
, 3: from^#(X) -> c_2(from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ MAYBE ]
->{2} [ NA ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {1},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {1},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(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]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {},
Uargs(from^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
from^#(x1) = [3 3] x1 + [0]
[3 3] [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: {from^#(X) -> c_2(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: first^#(0(), X) -> c_0()
, 2: first^#(s(X), cons(Y, Z)) -> c_1(first^#(X, Z))
, 3: from^#(X) -> c_2(from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ MAYBE ]
->{2} [ NA ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {1},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
first^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {1},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {},
Uargs(from^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
from^#(x1) = [3] 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: {from^#(X) -> c_2(from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
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:
{ first(0(), X) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
, from(X) -> cons(X, from(s(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: first^#(0(), X) -> c_0()
, 2: first^#(s(X), cons(Y, Z)) -> c_1(Y, first^#(X, Z))
, 3: from^#(X) -> c_2(X, from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ MAYBE ]
->{2} [ NA ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {2},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(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]
nil() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
[0 1 3] [0 1 0] [0]
[0 0 1] [0 0 0] [0]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(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]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {2},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(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]
nil() = [0]
[0]
[0]
s(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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
from^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(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]
We have not generated a proof for the resulting sub-problem.
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {},
Uargs(from^#) = {}, Uargs(c_2) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(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]
nil() = [0]
[0]
[0]
s(x1) = [0 1 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [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]
from(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
first^#(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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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_2(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]
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_2(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: first^#(0(), X) -> c_0()
, 2: first^#(s(X), cons(Y, Z)) -> c_1(Y, first^#(X, Z))
, 3: from^#(X) -> c_2(X, from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ MAYBE ]
->{2} [ NA ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {2},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
[0 1] [0 1] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {2},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
We have not generated a proof for the resulting sub-problem.
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {},
Uargs(from^#) = {}, Uargs(c_2) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from(x1) = [0 0] x1 + [0]
[0 0] [0]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
from^#(x1) = [1 3] x1 + [0]
[3 3] [0]
c_2(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 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: {from^#(X) -> c_2(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: first^#(0(), X) -> c_0()
, 2: first^#(s(X), cons(Y, Z)) -> c_1(Y, first^#(X, Z))
, 3: from^#(X) -> c_2(X, from^#(s(X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ MAYBE ]
->{2} [ NA ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {2},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
from(x1) = [0] x1 + [0]
first^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [1] x1 + [1] x2 + [0]
from^#(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {2}->{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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {2},
Uargs(from^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [1] x2 + [0]
from^#(x1) = [0] x1 + [0]
c_2(x1, x2) = [0] x1 + [0] x2 + [0]
We have not generated a proof for the resulting sub-problem.
* Path {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(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(from) = {}, Uargs(first^#) = {}, Uargs(c_1) = {},
Uargs(from^#) = {}, Uargs(c_2) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1, x2) = [0] x1 + [0] x2 + [0]
from^#(x1) = [3] x1 + [0]
c_2(x1, x2) = [2] x1 + [1] x2 + [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_2(X, from^#(s(X)))}
Weak Rules: {}
Proof Output:
The input cannot be shown compatible
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.