Tool CaT
stdout:
MAYBE
Problem:
f(tt(),x) -> f(isList(x),x)
isList(Cons(x,xs())) -> isList(xs())
isList(nil()) -> tt()
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(tt(), x) -> f(isList(x), x)
, isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
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^#(tt(), x) -> c_0(f^#(isList(x), x))
, 2: isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))
, 3: isList^#(nil()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f^#(tt(), x) -> c_0(f^#(isList(x), x))
, isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_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]
tt() = [0]
[0]
[0]
isList(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]
xs() = [0]
[0]
[0]
nil() = [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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [1]
[0 0 0] [0 0 1] [2]
[0 0 0] [0 0 0] [0]
xs() = [0]
[2]
[2]
isList^#(x1) = [2 3 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [2 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
* Path {3}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_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]
tt() = [0]
[0]
[0]
isList(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]
xs() = [0]
[0]
[0]
nil() = [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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isList^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
isList^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_2() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(tt(), x) -> c_0(f^#(isList(x), x))
, 2: isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))
, 3: isList^#(nil()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f^#(tt(), x) -> c_0(f^#(isList(x), x))
, isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_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]
tt() = [0]
[0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
xs() = [0]
[0]
nil() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [0]
xs() = [0]
[0]
isList^#(x1) = [2 0] x1 + [4]
[2 0] [2]
c_1(x1) = [0 0] x1 + [7]
[0 2] [1]
* Path {3}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_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]
tt() = [0]
[0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
xs() = [0]
[0]
nil() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isList^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
isList^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(tt(), x) -> c_0(f^#(isList(x), x))
, 2: isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))
, 3: isList^#(nil()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f^#(tt(), x) -> c_0(f^#(isList(x), x))
, isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
xs() = [0]
nil() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0] x1 + [0] x2 + [2]
xs() = [0]
isList^#(x1) = [2] x1 + [2]
c_1(x1) = [2] x1 + [1]
* Path {3}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
xs() = [0]
nil() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isList^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
isList^#(x1) = [1] x1 + [7]
c_2() = [1]
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(tt(), x) -> f(isList(x), x)
, isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
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^#(tt(), x) -> c_0(f^#(isList(x), x))
, 2: isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))
, 3: isList^#(nil()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f^#(tt(), x) -> c_0(f^#(isList(x), x))
, isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_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]
tt() = [0]
[0]
[0]
isList(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]
xs() = [0]
[0]
[0]
nil() = [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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [1]
[0 0 0] [0 0 1] [2]
[0 0 0] [0 0 0] [0]
xs() = [0]
[2]
[2]
isList^#(x1) = [2 3 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [2 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
* Path {3}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_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]
tt() = [0]
[0]
[0]
isList(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]
xs() = [0]
[0]
[0]
nil() = [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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isList^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
isList^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_2() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(tt(), x) -> c_0(f^#(isList(x), x))
, 2: isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))
, 3: isList^#(nil()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f^#(tt(), x) -> c_0(f^#(isList(x), x))
, isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_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]
tt() = [0]
[0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
xs() = [0]
[0]
nil() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [0]
xs() = [0]
[0]
isList^#(x1) = [2 0] x1 + [4]
[2 0] [2]
c_1(x1) = [0 0] x1 + [7]
[0 2] [1]
* Path {3}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_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]
tt() = [0]
[0]
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
xs() = [0]
[0]
nil() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isList^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
isList^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(tt(), x) -> c_0(f^#(isList(x), x))
, 2: isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))
, 3: isList^#(nil()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f^#(tt(), x) -> c_0(f^#(isList(x), x))
, isList(Cons(x, xs())) -> isList(xs())
, isList(nil()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
xs() = [0]
nil() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(Cons(x, xs())) -> c_1(isList^#(xs()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0] x1 + [0] x2 + [2]
xs() = [0]
isList^#(x1) = [2] x1 + [2]
c_1(x1) = [2] x1 + [1]
* Path {3}: YES(?,O(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(f) = {}, Uargs(isList) = {}, Uargs(Cons) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(isList^#) = {},
Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
isList(x1) = [0] x1 + [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
xs() = [0]
nil() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
isList^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(isList^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
isList^#(x1) = [1] x1 + [7]
c_2() = [1]
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.