Tool CaT
stdout:
MAYBE
Problem:
f(g(x),s(0())) -> f(g(x),g(x))
g(s(x)) -> s(g(x))
g(0()) -> 0()
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(g(x), s(0())) -> f(g(x), g(x))
, g(s(x)) -> s(g(x))
, g(0()) -> 0()}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(g(x), s(0())) -> f(g(x), g(x))
, g(s(x)) -> s(g(x))
, g(0()) -> 0()}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(x), s(0())) -> c_0(f^#(g(x), g(x)))
, 2: g^#(s(x)) -> c_1(g^#(x))
, 3: g^#(0()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(n^1))
-----------------------
The usable rules for this path are:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {1}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [2] x1 + [1]
s(x1) = [1] x1 + [2]
0() = [3]
f^#(x1, x2) = [2] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(g(x), s(0())) -> c_0(f^#(g(x), g(x)))}
Weak Rules:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0()}
Proof Output:
The following argument positions are usable:
Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [2] x1 + [1]
s(x1) = [0] x1 + [4]
0() = [0]
f^#(x1, x2) = [2] x1 + [0] x2 + [6]
c_0(x1) = [0] x1 + [7]
* 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(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [3] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [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: {g^#(s(x)) -> c_1(g^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
g^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [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: {g^#(0()) -> c_2()}
Weak Rules: {g^#(s(x)) -> c_1(g^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
g^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [1]Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ f(g(x), s(0())) -> f(g(x), g(x))
, g(s(x)) -> s(g(x))
, g(0()) -> 0()}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ f(g(x), s(0())) -> f(g(x), g(x))
, g(s(x)) -> s(g(x))
, g(0()) -> 0()}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(x), s(0())) -> c_0(f^#(g(x), g(x)))
, 2: g^#(s(x)) -> c_1(g^#(x))
, 3: g^#(0()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(n^1))
-----------------------
The usable rules for this path are:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {1}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [2] x1 + [1]
s(x1) = [1] x1 + [2]
0() = [3]
f^#(x1, x2) = [2] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(g(x), s(0())) -> c_0(f^#(g(x), g(x)))}
Weak Rules:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0()}
Proof Output:
The following argument positions are usable:
Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1) = [2] x1 + [1]
s(x1) = [0] x1 + [4]
0() = [0]
f^#(x1, x2) = [2] x1 + [0] x2 + [6]
c_0(x1) = [0] x1 + [7]
* 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(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [3] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [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: {g^#(s(x)) -> c_1(g^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
g^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [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: {g^#(0()) -> c_2()}
Weak Rules: {g^#(s(x)) -> c_1(g^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
g^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [1]