Tool CaT
stdout:
MAYBE
Problem:
div(x,s(y)) -> d(x,s(y),0())
d(x,s(y),z) -> cond(ge(x,z),x,y,z)
cond(true(),x,y,z) -> s(d(x,s(y),plus(s(y),z)))
cond(false(),x,y,z) -> 0()
ge(u,0()) -> true()
ge(0(),s(v)) -> false()
ge(s(u),s(v)) -> ge(u,v)
plus(n,0()) -> n
plus(n,s(m)) -> s(plus(n,m))
plus(plus(n,m),u) -> plus(n,plus(m,u))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ div(x, s(y)) -> d(x, s(y), 0())
, d(x, s(y), z) -> cond(ge(x, z), x, y, z)
, cond(true(), x, y, z) -> s(d(x, s(y), plus(s(y), z)))
, cond(false(), x, y, z) -> 0()
, ge(u, 0()) -> true()
, ge(0(), s(v)) -> false()
, ge(s(u), s(v)) -> ge(u, v)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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: div^#(x, s(y)) -> c_0(d^#(x, s(y), 0()))
, 2: d^#(x, s(y), z) -> c_1(cond^#(ge(x, z), x, y, z))
, 3: cond^#(true(), x, y, z) -> c_2(d^#(x, s(y), plus(s(y), z)))
, 4: cond^#(false(), x, y, z) -> c_3()
, 5: ge^#(u, 0()) -> c_4()
, 6: ge^#(0(), s(v)) -> c_5()
, 7: ge^#(s(u), s(v)) -> c_6(ge^#(u, v))
, 8: plus^#(n, 0()) -> c_7()
, 9: plus^#(n, s(m)) -> c_8(plus^#(n, m))
, 10: plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9,10} [ inherited ]
|
`->{8} [ MAYBE ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^3)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2,3} [ inherited ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(u, 0()) -> true()
, ge(0(), s(v)) -> false()
, ge(s(u), s(v)) -> ge(u, v)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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 {7}: 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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
cond(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
ge^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_6(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {7}->{5}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
cond(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(u, 0()) -> c_4()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
0() = [2]
[2]
[2]
ge^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_4() = [1]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {7}->{6}: 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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
cond(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(v)) -> c_5()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
0() = [0]
[2]
[0]
ge^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_5() = [1]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {9,10}: inherited
----------------------
This path is subsumed by the proof of path {9,10}->{8}.
* Path {9,10}->{8}: MAYBE
-----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus^#(n, s(m)) -> c_8(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_7()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(x, s(y)) -> c_0(d^#(x, s(y), 0()))
, 2: d^#(x, s(y), z) -> c_1(cond^#(ge(x, z), x, y, z))
, 3: cond^#(true(), x, y, z) -> c_2(d^#(x, s(y), plus(s(y), z)))
, 4: cond^#(false(), x, y, z) -> c_3()
, 5: ge^#(u, 0()) -> c_4()
, 6: ge^#(0(), s(v)) -> c_5()
, 7: ge^#(s(u), s(v)) -> c_6(ge^#(u, v))
, 8: plus^#(n, 0()) -> c_7()
, 9: plus^#(n, s(m)) -> c_8(plus^#(n, m))
, 10: plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9,10} [ inherited ]
|
`->{8} [ MAYBE ]
->{7} [ YES(?,O(n^2)) ]
|
|->{5} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2,3} [ inherited ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(u, 0()) -> true()
, ge(0(), s(v)) -> false()
, ge(s(u), s(v)) -> ge(u, v)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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 {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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
d(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
cond(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
ge^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [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: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
ge^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_6(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {7}->{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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
cond(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [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: {ge^#(u, 0()) -> c_4()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 1] [0]
0() = [2]
[0]
ge^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_4() = [1]
[0]
c_6(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {7}->{6}: 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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
cond(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [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: {ge^#(0(), s(v)) -> c_5()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [2]
0() = [2]
[2]
ge^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_5() = [1]
[0]
c_6(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {9,10}: inherited
----------------------
This path is subsumed by the proof of path {9,10}->{8}.
* Path {9,10}->{8}: MAYBE
-----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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:
{ plus^#(n, s(m)) -> c_8(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_7()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(x, s(y)) -> c_0(d^#(x, s(y), 0()))
, 2: d^#(x, s(y), z) -> c_1(cond^#(ge(x, z), x, y, z))
, 3: cond^#(true(), x, y, z) -> c_2(d^#(x, s(y), plus(s(y), z)))
, 4: cond^#(false(), x, y, z) -> c_3()
, 5: ge^#(u, 0()) -> c_4()
, 6: ge^#(0(), s(v)) -> c_5()
, 7: ge^#(s(u), s(v)) -> c_6(ge^#(u, v))
, 8: plus^#(n, 0()) -> c_7()
, 9: plus^#(n, s(m)) -> c_8(plus^#(n, m))
, 10: plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9,10} [ inherited ]
|
`->{8} [ MAYBE ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
`->{2,3} [ inherited ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(u, 0()) -> true()
, ge(0(), s(v)) -> false()
, ge(s(u), s(v)) -> ge(u, v)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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 {7}: 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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
d(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
cond(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
cond^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
ge^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(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: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
d(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
cond(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
cond^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(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: {ge^#(u, 0()) -> c_4()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_4() = [1]
c_6(x1) = [1] x1 + [7]
* Path {7}->{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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
d(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
cond(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
cond^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
c_9(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: {ge^#(0(), s(v)) -> c_5()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5() = [1]
c_6(x1) = [1] x1 + [7]
* Path {9,10}: inherited
----------------------
This path is subsumed by the proof of path {9,10}->{8}.
* Path {9,10}->{8}: MAYBE
-----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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:
{ plus^#(n, s(m)) -> c_8(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_7()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) '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
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ div(x, s(y)) -> d(x, s(y), 0())
, d(x, s(y), z) -> cond(ge(x, z), x, y, z)
, cond(true(), x, y, z) -> s(d(x, s(y), plus(s(y), z)))
, cond(false(), x, y, z) -> 0()
, ge(u, 0()) -> true()
, ge(0(), s(v)) -> false()
, ge(s(u), s(v)) -> ge(u, v)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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: div^#(x, s(y)) -> c_0(d^#(x, s(y), 0()))
, 2: d^#(x, s(y), z) -> c_1(cond^#(ge(x, z), x, y, z))
, 3: cond^#(true(), x, y, z) -> c_2(d^#(x, s(y), plus(s(y), z)))
, 4: cond^#(false(), x, y, z) -> c_3()
, 5: ge^#(u, 0()) -> c_4()
, 6: ge^#(0(), s(v)) -> c_5()
, 7: ge^#(s(u), s(v)) -> c_6(ge^#(u, v))
, 8: plus^#(n, 0()) -> c_7(n)
, 9: plus^#(n, s(m)) -> c_8(plus^#(n, m))
, 10: plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9,10} [ inherited ]
|
`->{8} [ MAYBE ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^3)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2,3} [ inherited ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(u, 0()) -> true()
, ge(0(), s(v)) -> false()
, ge(s(u), s(v)) -> ge(u, v)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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 {7}: 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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
cond(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
ge^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_6(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {7}->{5}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
cond(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(u, 0()) -> c_4()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
0() = [2]
[2]
[2]
ge^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_4() = [1]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {7}->{6}: 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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
cond(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
ge(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
div^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
d^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
ge^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(v)) -> c_5()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
0() = [0]
[2]
[0]
ge^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_5() = [1]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {9,10}: inherited
----------------------
This path is subsumed by the proof of path {9,10}->{8}.
* Path {9,10}->{8}: MAYBE
-----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ plus^#(n, s(m)) -> c_8(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_7(n)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(x, s(y)) -> c_0(d^#(x, s(y), 0()))
, 2: d^#(x, s(y), z) -> c_1(cond^#(ge(x, z), x, y, z))
, 3: cond^#(true(), x, y, z) -> c_2(d^#(x, s(y), plus(s(y), z)))
, 4: cond^#(false(), x, y, z) -> c_3()
, 5: ge^#(u, 0()) -> c_4()
, 6: ge^#(0(), s(v)) -> c_5()
, 7: ge^#(s(u), s(v)) -> c_6(ge^#(u, v))
, 8: plus^#(n, 0()) -> c_7(n)
, 9: plus^#(n, s(m)) -> c_8(plus^#(n, m))
, 10: plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9,10} [ inherited ]
|
`->{8} [ MAYBE ]
->{7} [ YES(?,O(n^2)) ]
|
|->{5} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{1} [ inherited ]
|
`->{2,3} [ inherited ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(u, 0()) -> true()
, ge(0(), s(v)) -> false()
, ge(s(u), s(v)) -> ge(u, v)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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 {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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
d(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
cond(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
ge^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [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: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
ge^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_6(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {7}->{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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
cond(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [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: {ge^#(u, 0()) -> c_4()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 1] [0]
0() = [2]
[0]
ge^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_4() = [1]
[0]
c_6(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {7}->{6}: 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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
d(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
cond(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
d^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [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: {ge^#(0(), s(v)) -> c_5()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [2]
0() = [2]
[2]
ge^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_5() = [1]
[0]
c_6(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {9,10}: inherited
----------------------
This path is subsumed by the proof of path {9,10}->{8}.
* Path {9,10}->{8}: MAYBE
-----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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: runtime-complexity with respect to
Rules:
{ plus^#(n, s(m)) -> c_8(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_7(n)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: div^#(x, s(y)) -> c_0(d^#(x, s(y), 0()))
, 2: d^#(x, s(y), z) -> c_1(cond^#(ge(x, z), x, y, z))
, 3: cond^#(true(), x, y, z) -> c_2(d^#(x, s(y), plus(s(y), z)))
, 4: cond^#(false(), x, y, z) -> c_3()
, 5: ge^#(u, 0()) -> c_4()
, 6: ge^#(0(), s(v)) -> c_5()
, 7: ge^#(s(u), s(v)) -> c_6(ge^#(u, v))
, 8: plus^#(n, 0()) -> c_7(n)
, 9: plus^#(n, s(m)) -> c_8(plus^#(n, m))
, 10: plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{9,10} [ inherited ]
|
`->{8} [ MAYBE ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{1} [ inherited ]
|
`->{2,3} [ inherited ]
|
`->{4} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}: inherited
--------------------------
This path is subsumed by the proof of path {1}->{2,3}->{4}.
* Path {1}->{2,3}->{4}: NA
------------------------
The usable rules for this path are:
{ ge(u, 0()) -> true()
, ge(0(), s(v)) -> false()
, ge(s(u), s(v)) -> ge(u, v)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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 {7}: 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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
d(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
cond(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
cond^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
ge^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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: DP runtime-complexity with respect to
Strict Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
d(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
cond(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
cond^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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: DP runtime-complexity with respect to
Strict Rules: {ge^#(u, 0()) -> c_4()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_4() = [1]
c_6(x1) = [1] x1 + [7]
* Path {7}->{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(div) = {}, Uargs(s) = {}, Uargs(d) = {}, Uargs(cond) = {},
Uargs(ge) = {}, Uargs(plus) = {}, Uargs(div^#) = {},
Uargs(c_0) = {}, Uargs(d^#) = {}, Uargs(c_1) = {},
Uargs(cond^#) = {}, Uargs(c_2) = {}, Uargs(ge^#) = {},
Uargs(c_6) = {1}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
div(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
d(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
cond(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
d^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
cond^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9(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: DP runtime-complexity with respect to
Strict Rules: {ge^#(0(), s(v)) -> c_5()}
Weak Rules: {ge^#(s(u), s(v)) -> c_6(ge^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5() = [1]
c_6(x1) = [1] x1 + [7]
* Path {9,10}: inherited
----------------------
This path is subsumed by the proof of path {9,10}->{8}.
* Path {9,10}->{8}: MAYBE
-----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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: runtime-complexity with respect to
Rules:
{ plus^#(n, s(m)) -> c_8(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_9(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_7(n)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.