Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.41 |
|---|
stdout:
MAYBE
Problem:
p(s(x)) -> x
fac(0()) -> s(0())
fac(s(x)) -> times(s(x),fac(p(s(x))))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^1))
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^3)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^3))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^3))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: p^#(s(x)) -> c_0()
, 2: fac^#(0()) -> c_1()
, 3: fac^#(s(x)) -> c_2(fac^#(p(s(x))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^3)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(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(p) = {}, Uargs(s) = {}, Uargs(fac) = {}, Uargs(times) = {},
Uargs(p^#) = {}, Uargs(fac^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
times(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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
fac^#(x1) = [0 0 0] x1 + [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]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
p^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {}, Uargs(times) = {},
Uargs(p^#) = {}, Uargs(fac^#) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [3]
[0 1 0] [3]
[0 0 1] [3]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
times(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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
fac^#(x1) = [2 1 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {fac^#(s(x)) -> c_2(fac^#(p(s(x))))}
Weak Rules: {p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [2 0 0] x1 + [0]
[1 0 0] [0]
[0 1 0] [0]
s(x1) = [1 4 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [4]
fac^#(x1) = [0 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [0]
* Path {3}->{2}: YES(?,O(n^3))
----------------------------
The usable rules for this path are:
{p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {}, Uargs(times) = {},
Uargs(p^#) = {}, Uargs(fac^#) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [3 3 3] x1 + [3]
[0 2 0] [3]
[0 0 1] [3]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
times(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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
fac^#(x1) = [3 0 0] x1 + [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]
Complexity induced by the adequate RMI: YES(?,O(n^3))
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: {fac^#(0()) -> c_1()}
Weak Rules:
{ fac^#(s(x)) -> c_2(fac^#(p(s(x))))
, p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 4 0] [0]
s(x1) = [1 3 0] x1 + [0]
[0 1 2] [0]
[0 0 0] [0]
0() = [2]
[0]
[0]
fac^#(x1) = [2 2 0] x1 + [0]
[0 4 0] [4]
[0 0 0] [0]
c_1() = [1]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [3]
[0 0 0] [0]Tool RC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.41 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^3)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^3))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^3))
Input Problem: runtime-complexity with respect to
Rules:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^3))
Input Problem: runtime-complexity with respect to
Rules:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: p^#(s(x)) -> c_0(x)
, 2: fac^#(0()) -> c_1()
, 3: fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^3)) ]
->{1} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {}, Uargs(times) = {},
Uargs(p^#) = {}, Uargs(c_0) = {}, Uargs(fac^#) = {},
Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 3] x1 + [0]
[0 1 1] [0]
[0 0 1] [0]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
times(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]
p^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
fac^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(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]
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: {p^#(s(x)) -> c_0(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_0) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
p^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {}, Uargs(times) = {},
Uargs(p^#) = {}, Uargs(c_0) = {}, Uargs(fac^#) = {1},
Uargs(c_2) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [3]
[0 1 0] [3]
[0 0 1] [3]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
times(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]
p^#(x1) = [0 0 0] x1 + [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]
fac^#(x1) = [2 1 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2(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]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: {fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))}
Weak Rules: {p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac^#) = {}, Uargs(c_2) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [0]
[2 0 0] [0]
[0 1 0] [0]
s(x1) = [1 1 0] x1 + [1]
[0 0 1] [0]
[0 0 1] [1]
fac^#(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
* Path {3}->{2}: YES(?,O(n^3))
----------------------------
The usable rules for this path are:
{p(s(x)) -> x}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {}, Uargs(times) = {},
Uargs(p^#) = {}, Uargs(c_0) = {}, Uargs(fac^#) = {1},
Uargs(c_2) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [3 3 3] x1 + [3]
[0 2 0] [3]
[0 0 1] [3]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
fac(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
times(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]
p^#(x1) = [0 0 0] x1 + [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]
fac^#(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(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]
Complexity induced by the adequate RMI: YES(?,O(n^3))
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: {fac^#(0()) -> c_1()}
Weak Rules:
{ fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
, p(s(x)) -> x}
Proof Output:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac^#) = {}, Uargs(c_2) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [0]
[2 0 0] [0]
[0 4 0] [0]
s(x1) = [1 4 0] x1 + [0]
[0 0 1] [0]
[0 0 0] [0]
0() = [2]
[0]
[0]
fac^#(x1) = [2 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [1]
[0]
[0]
c_2(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^2))
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [1]
[1 0 0] [0]
[0 1 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
fac(x1) = [1 0 2] x1 + [0]
[0 0 0] [1]
[2 0 0] [0]
0() = [2]
[0]
[1]
times(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [0]
Hurray, we answered YES(?,O(n^2))Tool pair2rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^2))
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [1]
[1 0 0] [0]
[0 1 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
fac(x1) = [1 0 2] x1 + [0]
[0 0 0] [1]
[2 0 0] [0]
0() = [2]
[0]
[1]
times(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [0]
Hurray, we answered YES(?,O(n^2))Tool pair3irc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
The input problem contains no overlaps that give rise to inapplicable rules.
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: innermost
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [1]
[1 0 0] [0]
[0 1 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
fac(x1) = [1 0 2] x1 + [0]
[0 0 0] [1]
[2 0 0] [0]
0() = [2]
[0]
[1]
times(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [0]
Hurray, we answered YES(?,O(n^2))Tool pair3rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^2))
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [1]
[1 0 0] [0]
[0 1 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
fac(x1) = [1 0 2] x1 + [0]
[0 0 0] [1]
[2 0 0] [0]
0() = [2]
[0]
[1]
times(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [0]
Hurray, we answered YES(?,O(n^2))Tool rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^2))
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [1]
[1 0 0] [0]
[0 1 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
fac(x1) = [1 0 2] x1 + [0]
[0 0 0] [1]
[2 0 0] [0]
0() = [2]
[0]
[1]
times(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [0]
Hurray, we answered YES(?,O(n^2))Tool tup3irc
| Execution Time | 4.631448ms |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.41 |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
The input problem contains no overlaps that give rise to inapplicable rules.
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ p(s(x)) -> x
, fac(0()) -> s(0())
, fac(s(x)) -> times(s(x), fac(p(s(x))))}
StartTerms: basic terms
Strategy: innermost
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
p(x1) = [1 0 0] x1 + [1]
[1 0 0] [0]
[0 1 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
fac(x1) = [1 0 2] x1 + [0]
[0 0 0] [1]
[2 0 0] [0]
0() = [2]
[0]
[1]
times(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [1]
[0 0 0] [0 0 0] [0]
Hurray, we answered YES(?,O(n^2))