Tool CaT
stdout:
MAYBE
Problem:
f(empty(),l) -> l
f(cons(x,k),l) -> g(k,l,cons(x,k))
g(a,b,c) -> f(a,cons(b,c))
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(empty(), l) -> l
, f(cons(x, k), l) -> g(k, l, cons(x, k))
, g(a, b, c) -> f(a, cons(b, c))}
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(empty(), l) -> l
, f(cons(x, k), l) -> g(k, l, cons(x, k))
, g(a, b, c) -> f(a, cons(b, c))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(empty(), l) -> c_0()
, 2: f^#(cons(x, k), l) -> c_1(g^#(k, l, cons(x, k)))
, 3: g^#(a, b, c) -> c_2(f^#(a, cons(b, c)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* 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(cons) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
g^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_2(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:
{ f^#(cons(x, k), l) -> c_1(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_2(f^#(a, cons(b, c)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [4]
f^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [1]
g^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [1]
c_2(x1) = [1] x1 + [0]
* Path {2,3}->{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(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(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: {f^#(empty(), l) -> c_0()}
Weak Rules:
{ f^#(cons(x, k), l) -> c_1(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_2(f^#(a, cons(b, c)))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [1]
c_2(x1) = [1] x1 + [0]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(empty(), l) -> l
, f(cons(x, k), l) -> g(k, l, cons(x, k))
, g(a, b, c) -> f(a, cons(b, c))}
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(empty(), l) -> l
, f(cons(x, k), l) -> g(k, l, cons(x, k))
, g(a, b, c) -> f(a, cons(b, c))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(empty(), l) -> c_0(l)
, 2: f^#(cons(x, k), l) -> c_1(g^#(k, l, cons(x, k)))
, 3: g^#(a, b, c) -> c_2(f^#(a, cons(b, c)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* 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(cons) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(g^#) = {},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
g^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_2(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:
{ f^#(cons(x, k), l) -> c_1(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_2(f^#(a, cons(b, c)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [4]
f^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [1]
g^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [1]
c_2(x1) = [1] x1 + [0]
* Path {2,3}->{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(f) = {}, Uargs(cons) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(g^#) = {},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
f^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_2(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: {f^#(empty(), l) -> c_0(l)}
Weak Rules:
{ f^#(cons(x, k), l) -> c_1(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_2(f^#(a, cons(b, c)))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(g^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty() = [2]
cons(x1, x2) = [1] x1 + [1] x2 + [4]
f^#(x1, x2) = [2] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [1]
c_1(x1) = [1] x1 + [7]
g^#(x1, x2, x3) = [2] x1 + [0] x2 + [0] x3 + [0]
c_2(x1) = [1] x1 + [0]