Tool CaT
stdout:
MAYBE
Problem:
minus(x,y) -> cond(gt(x,y),x,y)
cond(false(),x,y) -> 0()
cond(true(),x,y) -> s(minus(x,s(y)))
gt(0(),v) -> false()
gt(s(u),0()) -> true()
gt(s(u),s(v)) -> gt(u,v)
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:
{ minus(x, y) -> cond(gt(x, y), x, y)
, cond(false(), x, y) -> 0()
, cond(true(), x, y) -> s(minus(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
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: minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, 2: cond^#(false(), x, y) -> c_1()
, 3: cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, 4: gt^#(0(), v) -> c_3()
, 5: gt^#(s(u), 0()) -> c_4()
, 6: gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{1,3} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,3}: MAYBE
-----------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
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:
{ minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Proof Output:
The input cannot be shown compatible
* Path {1,3}->{2}: NA
-------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {1}, Uargs(gt^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [1]
[0]
[0]
0() = [0]
[0]
[0]
true() = [1]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
cond^#(x1, x2, x3) = [3 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() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
gt^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(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]
false() = [0]
[0]
[0]
0() = [0]
[0]
[0]
true() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(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: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
gt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_5(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {6}->{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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(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]
false() = [0]
[0]
[0]
0() = [0]
[0]
[0]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(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: {gt^#(0(), v) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
gt^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 0] [0 4 0] [0]
c_3() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {6}->{5}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(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]
false() = [0]
[0]
[0]
0() = [0]
[0]
[0]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(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: {gt^#(s(u), 0()) -> c_4()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 2] [2]
[0 0 0] [2]
gt^#(x1, x2) = [0 2 2] x1 + [0 0 0] x2 + [0]
[0 0 0] [4 0 0] [4]
[0 0 0] [0 0 0] [2]
c_4() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[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: minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, 2: cond^#(false(), x, y) -> c_1()
, 3: cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, 4: gt^#(0(), v) -> c_3()
, 5: gt^#(s(u), 0()) -> c_4()
, 6: gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1,3} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,3}: MAYBE
-----------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
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:
{ minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Proof Output:
The input cannot be shown compatible
* Path {1,3}->{2}: NA
-------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {1}, Uargs(gt^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [3]
false() = [0]
[1]
0() = [0]
[0]
true() = [0]
[1]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
cond^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
minus^#(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() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(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: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_5(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {6}->{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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(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() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(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: {gt^#(0(), v) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_3() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}->{5}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(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() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(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^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_4()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [2]
gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [0]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, 2: cond^#(false(), x, y) -> c_1()
, 3: cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, 4: gt^#(0(), v) -> c_3()
, 5: gt^#(s(u), 0()) -> c_4()
, 6: gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1,3} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,3}: MAYBE
-----------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
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:
{ minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Proof Output:
The input cannot be shown compatible
* Path {1,3}->{2}: NA
-------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {1}, Uargs(gt^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [2] x2 + [1]
false() = [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
cond^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [0]
minus^#(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() = [0]
c_2(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(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: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
minus^#(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() = [0]
c_2(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(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: {gt^#(0(), v) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_3() = [1]
c_5(x1) = [1] x1 + [5]
* Path {6}->{5}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
minus^#(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() = [0]
c_2(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(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: {gt^#(s(u), 0()) -> c_4()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [1]
c_5(x1) = [1] x1 + [7]
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:
{ minus(x, y) -> cond(gt(x, y), x, y)
, cond(false(), x, y) -> 0()
, cond(true(), x, y) -> s(minus(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
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: minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, 2: cond^#(false(), x, y) -> c_1()
, 3: cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, 4: gt^#(0(), v) -> c_3()
, 5: gt^#(s(u), 0()) -> c_4()
, 6: gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{1,3} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,3}: MAYBE
-----------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
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:
{ minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Proof Output:
The input cannot be shown compatible
* Path {1,3}->{2}: NA
-------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {1}, Uargs(gt^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [1]
[0]
[0]
0() = [0]
[0]
[0]
true() = [1]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
cond^#(x1, x2, x3) = [3 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() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
gt^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(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]
false() = [0]
[0]
[0]
0() = [0]
[0]
[0]
true() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(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: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
gt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_5(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {6}->{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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(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]
false() = [0]
[0]
[0]
0() = [0]
[0]
[0]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(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: {gt^#(0(), v) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
gt^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 0] [0 4 0] [0]
c_3() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {6}->{5}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(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]
false() = [0]
[0]
[0]
0() = [0]
[0]
[0]
true() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(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() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(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: {gt^#(s(u), 0()) -> c_4()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 2] [2]
[0 0 0] [2]
gt^#(x1, x2) = [0 2 2] x1 + [0 0 0] x2 + [0]
[0 0 0] [4 0 0] [4]
[0 0 0] [0 0 0] [2]
c_4() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[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: minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, 2: cond^#(false(), x, y) -> c_1()
, 3: cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, 4: gt^#(0(), v) -> c_3()
, 5: gt^#(s(u), 0()) -> c_4()
, 6: gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1,3} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,3}: MAYBE
-----------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
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:
{ minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Proof Output:
The input cannot be shown compatible
* Path {1,3}->{2}: NA
-------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {1}, Uargs(gt^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [3]
false() = [0]
[1]
0() = [0]
[0]
true() = [0]
[1]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
cond^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
minus^#(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() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(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: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_5(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {6}->{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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(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() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(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: {gt^#(0(), v) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_3() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}->{5}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(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]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(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() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(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^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_4()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [2]
gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [0]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, 2: cond^#(false(), x, y) -> c_1()
, 3: cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, 4: gt^#(0(), v) -> c_3()
, 5: gt^#(s(u), 0()) -> c_4()
, 6: gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1,3} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,3}: MAYBE
-----------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
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:
{ minus^#(x, y) -> c_0(cond^#(gt(x, y), x, y))
, cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
Proof Output:
The input cannot be shown compatible
* Path {1,3}->{2}: NA
-------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
Uargs(c_2) = {1}, Uargs(gt^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [2] x2 + [1]
false() = [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
cond^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [0]
minus^#(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() = [0]
c_2(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(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: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
minus^#(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() = [0]
c_2(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(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: {gt^#(0(), v) -> c_3()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_3() = [1]
c_5(x1) = [1] x1 + [5]
* Path {6}->{5}: 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(minus) = {}, Uargs(cond) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(minus^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
Uargs(c_2) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
minus^#(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() = [0]
c_2(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(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: {gt^#(s(u), 0()) -> c_4()}
Weak Rules: {gt^#(s(u), s(v)) -> c_5(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4() = [1]
c_5(x1) = [1] x1 + [7]
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.