Tool CaT
Execution Time | Unknown |
---|
Answer | MAYBE |
---|
Input | AG01 3.57 |
---|
stdout:
MAYBE
Problem:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
minus(minus(x,y),z) -> minus(x,plus(y,z))
app(nil(),k) -> k
app(l,nil()) -> l
app(cons(x,l),k) -> cons(x,app(l,k))
sum(cons(x,nil())) -> cons(x,nil())
sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
Proof:
OpenTool IRC1
Execution Time | Unknown |
---|
Answer | MAYBE |
---|
Input | AG01 3.57 |
---|
stdout:
MAYBE
Tool IRC2
Execution Time | Unknown |
---|
Answer | MAYBE |
---|
Input | AG01 3.57 |
---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: plus^#(0(), y) -> c_4()
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
, 7: minus^#(minus(x, y), z) -> c_6(minus^#(x, plus(y, z)))
, 8: app^#(nil(), k) -> c_7()
, 9: app^#(l, nil()) -> c_8()
, 10: app^#(cons(x, l), k) -> c_9(app^#(l, k))
, 11: sum^#(cons(x, nil())) -> c_10()
, 12: sum^#(cons(x, cons(y, l))) ->
c_11(sum^#(cons(plus(x, y), l)))
, 13: sum^#(app(l, cons(x, cons(y, k)))) ->
c_12(sum^#(app(l, sum(cons(x, cons(y, k))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ inherited ]
|
|->{11} [ NA ]
|
`->{12} [ inherited ]
|
`->{11} [ NA ]
->{10} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{4} [ MAYBE ]
|
`->{3} [ NA ]
->{2,7} [ NA ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2,7}: NA
--------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {1}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {2}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 0] x1 + [0 1] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 3] x1 + [1 0] x2 + [2]
[0 2] [3 3] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2,7}->{1}: NA
-------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {1}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {2}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [3 0] x1 + [3 3] x2 + [2]
[2 0] [3 3] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}: MAYBE
---------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {2}, Uargs(s) = {1}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [3 1] x1 + [3 0] x2 + [2]
[0 1] [0 0] [2]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [2 0] x1 + [1 0] x2 + [2]
[0 0] [3 3] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {plus^#(s(x), y) -> c_5(plus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
plus^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_5(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {6}->{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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {plus^#(0(), y) -> c_4()}
Weak Rules: {plus^#(s(x), y) -> c_5(plus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [4]
[0 1] [0]
plus^#(x1, x2) = [1 3] x1 + [0 0] x2 + [4]
[2 2] [4 4] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {10}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [2]
app^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[4 5] [4 4] [0]
c_9(x1) = [1 0] x1 + [1]
[0 0] [7]
* Path {10}->{8}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {app^#(nil(), k) -> c_7()}
Weak Rules: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[0]
cons(x1, x2) = [0 0] x1 + [1 3] x2 + [2]
[0 0] [0 0] [2]
app^#(x1, x2) = [3 0] x1 + [0 0] x2 + [4]
[2 2] [0 4] [0]
c_7() = [1]
[0]
c_9(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {10}->{9}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
sum(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
sum^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {app^#(l, nil()) -> c_8()}
Weak Rules: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [0]
[2]
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
app^#(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[2 2] [4 0] [0]
c_8() = [1]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [7]
* Path {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{11}.
* Path {13}->{11}: NA
-------------------
The usable rules for this path are:
{ app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{11}.
* Path {13}->{12}->{11}: NA
-------------------------
The usable rules for this path are:
{ app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: plus^#(0(), y) -> c_4()
, 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
, 7: minus^#(minus(x, y), z) -> c_6(minus^#(x, plus(y, z)))
, 8: app^#(nil(), k) -> c_7()
, 9: app^#(l, nil()) -> c_8()
, 10: app^#(cons(x, l), k) -> c_9(app^#(l, k))
, 11: sum^#(cons(x, nil())) -> c_10()
, 12: sum^#(cons(x, cons(y, l))) ->
c_11(sum^#(cons(plus(x, y), l)))
, 13: sum^#(app(l, cons(x, cons(y, k)))) ->
c_12(sum^#(app(l, sum(cons(x, cons(y, k))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ inherited ]
|
|->{11} [ NA ]
|
`->{12} [ inherited ]
|
`->{11} [ NA ]
->{10} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{4} [ MAYBE ]
|
`->{3} [ NA ]
->{2,7} [ NA ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2,7}: NA
--------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {1}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {2}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [3] x1 + [3] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [2]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [2] x1 + [1] x2 + [1]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
minus^#(x1, x2) = [2] x1 + [1] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {2,7}->{1}: NA
-------------------
The usable rules for this path are:
{ plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {1}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {2}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [1]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [2] x1 + [3] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}: MAYBE
---------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {2}, Uargs(s) = {1}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2] x1 + [1] x2 + [1]
0() = [3]
s(x1) = [1] x1 + [2]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [2] x1 + [1] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {6}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {plus^#(s(x), y) -> c_5(plus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
plus^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{5}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {1}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {plus^#(0(), y) -> c_4()}
Weak Rules: {plus^#(s(x), y) -> c_5(plus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
plus^#(x1, x2) = [6] x1 + [7] x2 + [0]
c_4() = [1]
c_5(x1) = [1] x1 + [7]
* Path {10}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [4]
app^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{8}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {app^#(nil(), k) -> c_7()}
Weak Rules: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
cons(x1, x2) = [0] x1 + [1] x2 + [2]
app^#(x1, x2) = [2] x1 + [7] x2 + [4]
c_7() = [1]
c_9(x1) = [1] x1 + [4]
* Path {10}->{9}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(plus) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(sum) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_9) = {1}, Uargs(sum^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
sum(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
sum^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {app^#(l, nil()) -> c_8()}
Weak Rules: {app^#(cons(x, l), k) -> c_9(app^#(l, k))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(app^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
cons(x1, x2) = [0] x1 + [1] x2 + [2]
app^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_8() = [1]
c_9(x1) = [1] x1 + [4]
* Path {13}: inherited
--------------------
This path is subsumed by the proof of path {13}->{12}->{11}.
* Path {13}->{11}: NA
-------------------
The usable rules for this path are:
{ app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {13}->{12}: inherited
--------------------------
This path is subsumed by the proof of path {13}->{12}->{11}.
* Path {13}->{12}->{11}: NA
-------------------------
The usable rules for this path are:
{ app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
Execution Time | Unknown |
---|
Answer | MAYBE |
---|
Input | AG01 3.57 |
---|
stdout:
MAYBE
Tool RC2
Execution Time | Unknown |
---|
Answer | TIMEOUT |
---|
Input | AG01 3.57 |
---|
stdout:
TIMEOUT
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: TIMEOUT
Input Problem: runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))}
Proof Output:
Computation stopped due to timeout after 60.0 secondsTool pair1rc
Execution Time | Unknown |
---|
Answer | TIMEOUT |
---|
Input | AG01 3.57 |
---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
Execution Time | Unknown |
---|
Answer | TIMEOUT |
---|
Input | AG01 3.57 |
---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
Execution Time | Unknown |
---|
Answer | TIMEOUT |
---|
Input | AG01 3.57 |
---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3rc
Execution Time | Unknown |
---|
Answer | TIMEOUT |
---|
Input | AG01 3.57 |
---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
Execution Time | Unknown |
---|
Answer | TIMEOUT |
---|
Input | AG01 3.57 |
---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
Execution Time | 60.1041ms |
---|
Answer | TIMEOUT |
---|
Input | AG01 3.57 |
---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, minus(minus(x, y), z) -> minus(x, plus(y, z))
, app(nil(), k) -> k
, app(l, nil()) -> l
, app(cons(x, l), k) -> cons(x, app(l, k))
, sum(cons(x, nil())) -> cons(x, nil())
, sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
, sum(app(l, cons(x, cons(y, k)))) ->
sum(app(l, sum(cons(x, cons(y, k)))))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..