Tool CaT
stdout:
MAYBE
Problem:
f(x,y) -> cond(lt(x,y),x,y)
cond(tt(),x,y) -> f(s(x),s(y))
lt(0(),y) -> tt()
lt(s(x),s(y)) -> lt(x,y)
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, y) -> cond(lt(x, y), x, y)
, cond(tt(), x, y) -> f(s(x), s(y))
, lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
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, y) -> c_0(cond^#(lt(x, y), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: lt^#(0(), y) -> c_2()
, 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
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^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {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]
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
lt(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]
s(x1) = [1 0 0] x1 + [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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [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: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
lt^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_3(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {4}->{3}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {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]
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
lt(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]
s(x1) = [0 0 0] x1 + [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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), y) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
0() = [2]
[2]
[2]
lt^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: lt^#(0(), y) -> c_2()
, 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
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^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {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]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [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]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_3(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {4}->{3}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {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]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [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]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), y) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 1] [0]
0() = [2]
[2]
lt^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: lt^#(0(), y) -> c_2()
, 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
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^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{3}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), y) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
lt^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_2() = [1]
c_3(x1) = [1] x1 + [2]
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, y) -> cond(lt(x, y), x, y)
, cond(tt(), x, y) -> f(s(x), s(y))
, lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
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, y) -> c_0(cond^#(lt(x, y), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: lt^#(0(), y) -> c_2()
, 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
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^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {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]
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
lt(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]
s(x1) = [1 0 0] x1 + [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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [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: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
lt^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_3(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {4}->{3}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {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]
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
lt(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]
s(x1) = [0 0 0] x1 + [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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
lt^#(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_2() = [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^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), y) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
0() = [2]
[2]
[2]
lt^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: lt^#(0(), y) -> c_2()
, 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
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^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {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]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [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]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_3(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {4}->{3}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {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]
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [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]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), y) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 1] [0]
0() = [2]
[2]
lt^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, 3: lt^#(0(), y) -> c_2()
, 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,2}: MAYBE
-----------------
The usable rules for this path are:
{ lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
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^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
, cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
, lt(0(), y) -> tt()
, lt(s(x), s(y)) -> lt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{3}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
lt(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {lt^#(0(), y) -> c_2()}
Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
lt^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_2() = [1]
c_3(x1) = [1] x1 + [2]
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.