Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.6a |
|---|
stdout:
MAYBE
Problem:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
gcd(0(),y) -> y
gcd(s(x),0()) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.6a |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.6a |
|---|
stdout:
YES(?,O(n^2))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(0(), y) -> c_0()
, 2: le^#(s(x), 0()) -> c_1()
, 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
, 4: minus^#(x, 0()) -> c_3()
, 5: minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
, 6: gcd^#(0(), y) -> c_5()
, 7: gcd^#(s(x), 0()) -> c_6()
, 8: gcd^#(s(x), s(y)) -> c_7(if_gcd^#(le(y, x), s(x), s(y)))
, 9: if_gcd^#(true(), s(x), s(y)) -> c_8(gcd^#(minus(x, y), s(y)))
, 10: if_gcd^#(false(), s(x), s(y)) ->
c_9(gcd^#(minus(y, x), s(x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,10,9} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{7} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1},
Uargs(minus^#) = {}, Uargs(c_4) = {}, Uargs(gcd^#) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_2(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1},
Uargs(minus^#) = {}, Uargs(c_4) = {}, Uargs(gcd^#) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(0(), y) -> c_0()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
le^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1},
Uargs(minus^#) = {}, Uargs(c_4) = {}, Uargs(gcd^#) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: {le^#(s(x), 0()) -> c_1()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [2]
le^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_4) = {1}, Uargs(gcd^#) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_4(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {5}->{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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_4) = {1}, Uargs(gcd^#) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_3()}
Weak Rules: {minus^#(s(x), s(y)) -> c_4(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 3] x1 + [2]
[0 1] [2]
minus^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
[0 2] [3 2] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {7}: 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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_4) = {}, Uargs(gcd^#) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gcd^#(s(x), 0()) -> c_6()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gcd^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [0 0] x1 + [2]
[0 0] [2]
gcd^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [3]
c_6() = [0]
[1]
* Path {8,10,9}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_4) = {}, Uargs(gcd^#) = {1},
Uargs(c_7) = {1}, Uargs(if_gcd^#) = {1}, Uargs(c_8) = {1},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 1] x2 + [1]
[0 0] [0 0] [3]
0() = [0]
[0]
true() = [0]
[1]
s(x1) = [1 2] x1 + [1]
[0 1] [1]
false() = [0]
[1]
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [3]
[0 1] [0 0] [3]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
if_gcd^#(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ gcd^#(s(x), s(y)) -> c_7(if_gcd^#(le(y, x), s(x), s(y)))
, if_gcd^#(false(), s(x), s(y)) -> c_9(gcd^#(minus(y, x), s(x)))
, if_gcd^#(true(), s(x), s(y)) -> c_8(gcd^#(minus(x, y), s(y)))}
Weak Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(gcd^#) = {}, Uargs(c_7) = {1}, Uargs(if_gcd^#) = {},
Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 1] [4]
false() = [0]
[0]
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 2] [0 0] [0]
gcd^#(x1, x2) = [4 1] x1 + [4 0] x2 + [1]
[0 0] [0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 2] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [4 1] x2 + [4 0] x3 + [0]
[2 0] [0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [1]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {8,10,9}->{6}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_4) = {}, Uargs(gcd^#) = {1},
Uargs(c_7) = {1}, Uargs(if_gcd^#) = {1}, Uargs(c_8) = {1},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [2 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
0() = [0]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
false() = [1]
[1]
minus(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
[0 2] [0 0] [3]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
if_gcd^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [1 0] x1 + [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 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gcd^#(0(), y) -> c_5()}
Weak Rules:
{ gcd^#(s(x), s(y)) -> c_7(if_gcd^#(le(y, x), s(x), s(y)))
, if_gcd^#(false(), s(x), s(y)) -> c_9(gcd^#(minus(y, x), s(x)))
, if_gcd^#(true(), s(x), s(y)) -> c_8(gcd^#(minus(x, y), s(y)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(gcd^#) = {}, Uargs(c_7) = {1}, Uargs(if_gcd^#) = {},
Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [4 0] x1 + [0 0] x2 + [0]
[1 1] [4 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [4]
c_5() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [3]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]Tool RC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.6a |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.6a |
|---|
stdout:
YES(?,O(n^2))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^2))
Input Problem: runtime-complexity with respect to
Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^2))
Input Problem: runtime-complexity with respect to
Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(0(), y) -> c_0()
, 2: le^#(s(x), 0()) -> c_1()
, 3: le^#(s(x), s(y)) -> c_2(le^#(x, y))
, 4: minus^#(x, 0()) -> c_3(x)
, 5: minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
, 6: gcd^#(0(), y) -> c_5(y)
, 7: gcd^#(s(x), 0()) -> c_6(x)
, 8: gcd^#(s(x), s(y)) -> c_7(if_gcd^#(le(y, x), s(x), s(y)))
, 9: if_gcd^#(true(), s(x), s(y)) -> c_8(gcd^#(minus(x, y), s(y)))
, 10: if_gcd^#(false(), s(x), s(y)) ->
c_9(gcd^#(minus(y, x), s(x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,10,9} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{7} [ YES(?,O(n^2)) ]
->{5} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1},
Uargs(minus^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(gcd^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_2(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1},
Uargs(minus^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(gcd^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: {le^#(0(), y) -> c_0()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
le^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1},
Uargs(minus^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(gcd^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: {le^#(s(x), 0()) -> c_1()}
Weak Rules: {le^#(s(x), s(y)) -> c_2(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [2]
le^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {1},
Uargs(gcd^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: {minus^#(s(x), s(y)) -> c_4(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {5}->{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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {1},
Uargs(gcd^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [1 1] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: {minus^#(x, 0()) -> c_3(x)}
Weak Rules: {minus^#(s(x), s(y)) -> c_4(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [2 6] x2 + [0]
[2 5] [5 3] [0]
c_3(x1) = [0 0] x1 + [1]
[0 1] [0]
c_4(x1) = [1 0] x1 + [3]
[0 0] [2]
* Path {7}: 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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(gcd^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(if_gcd^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 1] [0]
false() = [0]
[0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(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: {gcd^#(s(x), 0()) -> c_6(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gcd^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [2]
[0 0] [2]
gcd^#(x1, x2) = [2 2] x1 + [0 2] x2 + [3]
[2 2] [0 2] [3]
c_6(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {8,10,9}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(gcd^#) = {1}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(c_7) = {1}, Uargs(if_gcd^#) = {1}, Uargs(c_8) = {1},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 1] x2 + [1]
[0 0] [0 0] [3]
0() = [0]
[0]
true() = [0]
[1]
s(x1) = [1 2] x1 + [1]
[0 1] [1]
false() = [0]
[1]
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [3]
[0 1] [0 0] [3]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
if_gcd^#(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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:
{ gcd^#(s(x), s(y)) -> c_7(if_gcd^#(le(y, x), s(x), s(y)))
, if_gcd^#(false(), s(x), s(y)) -> c_9(gcd^#(minus(y, x), s(x)))
, if_gcd^#(true(), s(x), s(y)) -> c_8(gcd^#(minus(x, y), s(y)))}
Weak Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(gcd^#) = {}, Uargs(c_7) = {1}, Uargs(if_gcd^#) = {},
Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 1] [4]
false() = [0]
[0]
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 2] [0 0] [0]
gcd^#(x1, x2) = [4 1] x1 + [4 0] x2 + [1]
[0 0] [0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 2] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [4 1] x2 + [4 0] x3 + [0]
[2 0] [0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [1]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {8,10,9}->{6}: YES(?,O(n^1))
---------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {},
Uargs(if_gcd) = {}, Uargs(le^#) = {}, Uargs(c_2) = {},
Uargs(minus^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(gcd^#) = {1}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(c_7) = {1}, Uargs(if_gcd^#) = {1}, Uargs(c_8) = {1},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [2 0] x1 + [0 0] x2 + [2]
[3 3] [3 3] [3]
0() = [0]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
false() = [1]
[0]
minus(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
[0 2] [0 0] [3]
gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_gcd(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gcd^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 1] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
if_gcd^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [1 0] x1 + [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 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gcd^#(0(), y) -> c_5(y)}
Weak Rules:
{ gcd^#(s(x), s(y)) -> c_7(if_gcd^#(le(y, x), s(x), s(y)))
, if_gcd^#(false(), s(x), s(y)) -> c_9(gcd^#(minus(y, x), s(x)))
, if_gcd^#(true(), s(x), s(y)) -> c_8(gcd^#(minus(x, y), s(y)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {},
Uargs(gcd^#) = {}, Uargs(c_5) = {}, Uargs(c_7) = {1},
Uargs(if_gcd^#) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [4 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
false() = [0]
[0]
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[4 2] [4 0] [0]
gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
if_gcd^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.6a |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
Uargs(if_gcd) = {1}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0 0] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[2]
[0]
true() = [0]
[0]
[0]
s(x1) = [1 1 2] x1 + [1]
[0 0 0] [2]
[0 0 1] [2]
false() = [0]
[0]
[0]
minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
[1 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
gcd(x1, x2) = [1 0 2] x1 + [1 0 0] x2 + [1]
[0 0 1] [0 1 1] [0]
[0 0 1] [0 0 1] [0]
if_gcd(x1, x2, x3) = [2 0 0] x1 + [1 1 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
[0 0 0] [0 0 1] [0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool pair2rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.6a |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
Uargs(if_gcd) = {1}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0 0] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[2]
[0]
true() = [0]
[0]
[0]
s(x1) = [1 1 2] x1 + [1]
[0 0 0] [2]
[0 0 1] [2]
false() = [0]
[0]
[0]
minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
[1 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
gcd(x1, x2) = [1 0 2] x1 + [1 0 0] x2 + [1]
[0 0 1] [0 1 1] [0]
[0 0 1] [0 0 1] [0]
if_gcd(x1, x2, x3) = [2 0 0] x1 + [1 1 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
[0 0 0] [0 0 1] [0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool pair3irc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.6a |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
Uargs(if_gcd) = {1}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0 0] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[2]
[0]
true() = [0]
[0]
[0]
s(x1) = [1 1 2] x1 + [1]
[0 0 0] [2]
[0 0 1] [2]
false() = [0]
[0]
[0]
minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
[1 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
gcd(x1, x2) = [1 0 2] x1 + [1 0 0] x2 + [1]
[0 0 1] [0 1 1] [0]
[0 0 1] [0 0 1] [0]
if_gcd(x1, x2, x3) = [2 0 0] x1 + [1 1 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
[0 0 0] [0 0 1] [0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool pair3rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.6a |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
Uargs(if_gcd) = {1}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0 0] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[2]
[0]
true() = [0]
[0]
[0]
s(x1) = [1 1 2] x1 + [1]
[0 0 0] [2]
[0 0 1] [2]
false() = [0]
[0]
[0]
minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
[1 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
gcd(x1, x2) = [1 0 2] x1 + [1 0 0] x2 + [1]
[0 0 1] [0 1 1] [0]
[0 0 1] [0 0 1] [0]
if_gcd(x1, x2, x3) = [2 0 0] x1 + [1 1 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
[0 0 0] [0 0 1] [0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.6a |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
Uargs(if_gcd) = {1}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0 0] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[2]
[0]
true() = [0]
[0]
[0]
s(x1) = [1 1 2] x1 + [1]
[0 0 0] [2]
[0 0 1] [2]
false() = [0]
[0]
[0]
minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
[1 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
gcd(x1, x2) = [1 0 2] x1 + [1 0 0] x2 + [1]
[0 0 1] [0 1 1] [0]
[0 0 1] [0 0 1] [0]
if_gcd(x1, x2, x3) = [2 0 0] x1 + [1 1 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
[0 0 0] [0 0 1] [0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool tup3irc
| Execution Time | 4.191192ms |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | AG01 3.6a |
|---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, gcd(0(), y) -> y
, gcd(s(x), 0()) -> s(x)
, gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
, if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
, if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), 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(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
Uargs(if_gcd) = {1}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0 0] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[2]
[0]
true() = [0]
[0]
[0]
s(x1) = [1 1 2] x1 + [1]
[0 0 0] [2]
[0 0 1] [2]
false() = [0]
[0]
[0]
minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
[1 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
gcd(x1, x2) = [1 0 2] x1 + [1 0 0] x2 + [1]
[0 0 1] [0 1 1] [0]
[0 0 1] [0 0 1] [0]
if_gcd(x1, x2, x3) = [2 0 0] x1 + [1 1 0] x2 + [1 0 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
[0 0 0] [0 0 1] [0 0 1] [0]
Hurray, we answered YES(?,O(n^2))