Tool CaT
stdout:
MAYBE
Problem:
isList(nil()) -> tt()
isList(Cons(x,xs)) -> isList(xs)
downfrom(0()) -> nil()
downfrom(s(x)) -> Cons(s(x),downfrom(x))
f(x) -> cond(isList(downfrom(x)),s(x))
cond(tt(),x) -> f(x)
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:
{ isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))
, f(x) -> cond(isList(downfrom(x)), s(x))
, cond(tt(), x) -> f(x)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: isList^#(nil()) -> c_0()
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: downfrom^#(0()) -> c_2()
, 4: downfrom^#(s(x)) -> c_3(downfrom^#(x))
, 5: f^#(x) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, 6: cond^#(tt(), x) -> c_5(f^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,6} [ NA ]
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^2)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
tt() = [0]
[0]
[0]
Cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
downfrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond(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]
isList^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
downfrom^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
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) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 1 2] [2]
[0 0 0] [0 0 0] [0]
isList^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_1(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {2}->{1}: 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
tt() = [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]
downfrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond(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]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
downfrom^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_0()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
Cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 1] [1]
[0 0 0] [0 0 0] [0]
isList^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {1}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
tt() = [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]
downfrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond(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]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
downfrom^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[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]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {downfrom^#(s(x)) -> c_3(downfrom^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
downfrom^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_3(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {1}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
tt() = [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]
downfrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond(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]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
downfrom^#(x1) = [0 0 0] x1 + [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]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {downfrom^#(0()) -> c_2()}
Weak Rules: {downfrom^#(s(x)) -> c_3(downfrom^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
downfrom^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* Path {5,6}: NA
--------------
The usable rules for this path are:
{ isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: isList^#(nil()) -> c_0()
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: downfrom^#(0()) -> c_2()
, 4: downfrom^#(s(x)) -> c_3(downfrom^#(x))
, 5: f^#(x) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, 6: cond^#(tt(), x) -> c_5(f^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,6} [ MAYBE ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
tt() = [0]
[0]
Cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
downfrom(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isList^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
downfrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [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: {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) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
isList^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
tt() = [0]
[0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
downfrom(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
downfrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [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: {isList^#(nil()) -> c_0()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
Cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [3]
isList^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [5]
[2 0] [3]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {1}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
tt() = [0]
[0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
downfrom(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
downfrom^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [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: {downfrom^#(s(x)) -> c_3(downfrom^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
downfrom^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {1}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
tt() = [0]
[0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
downfrom(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
downfrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [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: {downfrom^#(0()) -> c_2()}
Weak Rules: {downfrom^#(s(x)) -> c_3(downfrom^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [3]
downfrom^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [5]
[2 0] [3]
* Path {5,6}: MAYBE
-----------------
The usable rules for this path are:
{ isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
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) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, cond^#(tt(), x) -> c_5(f^#(x))
, isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: isList^#(nil()) -> c_0()
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: downfrom^#(0()) -> c_2()
, 4: downfrom^#(s(x)) -> c_3(downfrom^#(x))
, 5: f^#(x) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, 6: cond^#(tt(), x) -> c_5(f^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,6} [ MAYBE ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0] x1 + [0]
nil() = [0]
tt() = [0]
Cons(x1, x2) = [1] x1 + [0] x2 + [0]
downfrom(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
cond(x1, x2) = [0] x1 + [0] x2 + [0]
isList^#(x1) = [3] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
downfrom^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] 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: {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) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0] x1 + [1] x2 + [4]
isList^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0] x1 + [0]
nil() = [0]
tt() = [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
downfrom(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
cond(x1, x2) = [0] x1 + [0] x2 + [0]
isList^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
downfrom^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] 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: {isList^#(nil()) -> c_0()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
Cons(x1, x2) = [0] x1 + [1] x2 + [0]
isList^#(x1) = [2] x1 + [0]
c_0() = [1]
c_1(x1) = [1] x1 + [0]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {1}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0] x1 + [0]
nil() = [0]
tt() = [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
downfrom(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
f(x1) = [0] x1 + [0]
cond(x1, x2) = [0] x1 + [0] x2 + [0]
isList^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
downfrom^#(x1) = [3] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] 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: {downfrom^#(s(x)) -> c_3(downfrom^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
downfrom^#(x1) = [2] x1 + [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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {1}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0] x1 + [0]
nil() = [0]
tt() = [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
downfrom(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
cond(x1, x2) = [0] x1 + [0] x2 + [0]
isList^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
downfrom^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] 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: {downfrom^#(0()) -> c_2()}
Weak Rules: {downfrom^#(s(x)) -> c_3(downfrom^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
downfrom^#(x1) = [2] x1 + [0]
c_2() = [1]
c_3(x1) = [1] x1 + [0]
* Path {5,6}: MAYBE
-----------------
The usable rules for this path are:
{ isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
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) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, cond^#(tt(), x) -> c_5(f^#(x))
, isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
Proof Output:
The input cannot be shown compatible
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))
, f(x) -> cond(isList(downfrom(x)), s(x))
, cond(tt(), x) -> f(x)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: isList^#(nil()) -> c_0()
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: downfrom^#(0()) -> c_2()
, 4: downfrom^#(s(x)) -> c_3(x, downfrom^#(x))
, 5: f^#(x) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, 6: cond^#(tt(), x) -> c_5(f^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,6} [ NA ]
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^2)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
tt() = [0]
[0]
[0]
Cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
downfrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond(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]
isList^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
downfrom^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(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]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
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) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 1 2] [2]
[0 0 0] [0 0 0] [0]
isList^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_1(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {2}->{1}: 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
tt() = [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]
downfrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond(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]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
downfrom^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(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]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {isList^#(nil()) -> c_0()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
Cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
[0 0 0] [0 1 1] [1]
[0 0 0] [0 0 0] [0]
isList^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {2}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
tt() = [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]
downfrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond(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]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
downfrom^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {downfrom^#(s(x)) -> c_3(x, downfrom^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 5 6] x1 + [4]
[0 1 2] [2]
[0 0 0] [4]
downfrom^#(x1) = [1 3 0] x1 + [0]
[0 0 2] [2]
[0 0 2] [0]
c_3(x1, x2) = [0 4 3] x1 + [1 2 2] x2 + [1]
[0 0 0] [0 0 0] [7]
[0 0 0] [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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {2}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
tt() = [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]
downfrom(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond(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]
isList^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
downfrom^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {downfrom^#(0()) -> c_2()}
Weak Rules: {downfrom^#(s(x)) -> c_3(x, downfrom^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 6 4] x1 + [2]
[0 1 2] [2]
[0 0 0] [2]
downfrom^#(x1) = [1 2 2] x1 + [0]
[0 2 2] [2]
[2 0 2] [0]
c_2() = [1]
[0]
[0]
c_3(x1, x2) = [0 1 1] x1 + [1 2 0] x2 + [3]
[0 2 3] [0 0 0] [6]
[2 3 3] [0 2 0] [4]
* Path {5,6}: NA
--------------
The usable rules for this path are:
{ isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: isList^#(nil()) -> c_0()
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: downfrom^#(0()) -> c_2()
, 4: downfrom^#(s(x)) -> c_3(x, downfrom^#(x))
, 5: f^#(x) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, 6: cond^#(tt(), x) -> c_5(f^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,6} [ MAYBE ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
tt() = [0]
[0]
Cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
downfrom(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isList^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
downfrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [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: {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) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
isList^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
tt() = [0]
[0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
downfrom(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
downfrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [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: {isList^#(nil()) -> c_0()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
Cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
[0 0] [0 0] [3]
isList^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [5]
[2 0] [3]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {2}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
tt() = [0]
[0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
downfrom(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
downfrom^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_2() = [0]
[0]
c_3(x1, x2) = [0 3] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [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: {downfrom^#(s(x)) -> c_3(x, downfrom^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [2]
downfrom^#(x1) = [1 2] x1 + [2]
[4 0] [0]
c_3(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[2 3] [2 0] [0]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {2}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
tt() = [0]
[0]
Cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
downfrom(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
isList^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
downfrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
f^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [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: {downfrom^#(0()) -> c_2()}
Weak Rules: {downfrom^#(s(x)) -> c_3(x, downfrom^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [3]
[0 1] [0]
downfrom^#(x1) = [1 3] x1 + [2]
[3 2] [0]
c_2() = [1]
[0]
c_3(x1, x2) = [0 0] x1 + [1 0] x2 + [3]
[1 1] [2 0] [3]
* Path {5,6}: MAYBE
-----------------
The usable rules for this path are:
{ isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
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) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, cond^#(tt(), x) -> c_5(f^#(x))
, isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: isList^#(nil()) -> c_0()
, 2: isList^#(Cons(x, xs)) -> c_1(isList^#(xs))
, 3: downfrom^#(0()) -> c_2()
, 4: downfrom^#(s(x)) -> c_3(x, downfrom^#(x))
, 5: f^#(x) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, 6: cond^#(tt(), x) -> c_5(f^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5,6} [ MAYBE ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0] x1 + [0]
nil() = [0]
tt() = [0]
Cons(x1, x2) = [1] x1 + [0] x2 + [0]
downfrom(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
cond(x1, x2) = [0] x1 + [0] x2 + [0]
isList^#(x1) = [3] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
downfrom^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] 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: {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) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
Cons(x1, x2) = [0] x1 + [1] x2 + [4]
isList^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{1}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {1}, Uargs(downfrom^#) = {},
Uargs(c_3) = {}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0] x1 + [0]
nil() = [0]
tt() = [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
downfrom(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
cond(x1, x2) = [0] x1 + [0] x2 + [0]
isList^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
downfrom^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] 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: {isList^#(nil()) -> c_0()}
Weak Rules: {isList^#(Cons(x, xs)) -> c_1(isList^#(xs))}
Proof Output:
The following argument positions are usable:
Uargs(Cons) = {}, Uargs(isList^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
Cons(x1, x2) = [0] x1 + [1] x2 + [0]
isList^#(x1) = [2] x1 + [0]
c_0() = [1]
c_1(x1) = [1] x1 + [0]
* 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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {2}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0] x1 + [0]
nil() = [0]
tt() = [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
downfrom(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
f(x1) = [0] x1 + [0]
cond(x1, x2) = [0] x1 + [0] x2 + [0]
isList^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
downfrom^#(x1) = [3] x1 + [0]
c_2() = [0]
c_3(x1, x2) = [0] x1 + [1] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] 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: {downfrom^#(s(x)) -> c_3(x, downfrom^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
downfrom^#(x1) = [2] x1 + [0]
c_3(x1, x2) = [0] x1 + [1] x2 + [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(isList) = {}, Uargs(Cons) = {}, Uargs(downfrom) = {},
Uargs(s) = {}, Uargs(f) = {}, Uargs(cond) = {},
Uargs(isList^#) = {}, Uargs(c_1) = {}, Uargs(downfrom^#) = {},
Uargs(c_3) = {2}, Uargs(f^#) = {}, Uargs(c_4) = {},
Uargs(cond^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
isList(x1) = [0] x1 + [0]
nil() = [0]
tt() = [0]
Cons(x1, x2) = [0] x1 + [0] x2 + [0]
downfrom(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
cond(x1, x2) = [0] x1 + [0] x2 + [0]
isList^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
downfrom^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1, x2) = [0] x1 + [1] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] 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: {downfrom^#(0()) -> c_2()}
Weak Rules: {downfrom^#(s(x)) -> c_3(x, downfrom^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(downfrom^#) = {}, Uargs(c_3) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
downfrom^#(x1) = [3] x1 + [2]
c_2() = [1]
c_3(x1, x2) = [0] x1 + [1] x2 + [4]
* Path {5,6}: MAYBE
-----------------
The usable rules for this path are:
{ isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
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) -> c_4(cond^#(isList(downfrom(x)), s(x)))
, cond^#(tt(), x) -> c_5(f^#(x))
, isList(nil()) -> tt()
, isList(Cons(x, xs)) -> isList(xs)
, downfrom(0()) -> nil()
, downfrom(s(x)) -> Cons(s(x), downfrom(x))}
Proof Output:
The input cannot be shown compatible
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.