Tool CaT
stdout:
MAYBE
Problem:
cond1(true(),x,y) -> cond2(gr(x,y),x,y)
cond2(true(),x,y) -> cond1(neq(x,0()),y,y)
cond2(false(),x,y) -> cond1(neq(x,0()),p(x),y)
gr(0(),x) -> false()
gr(s(x),0()) -> true()
gr(s(x),s(y)) -> gr(x,y)
neq(0(),0()) -> false()
neq(0(),s(x)) -> true()
neq(s(x),0()) -> true()
neq(s(x),s(y)) -> neq(x,y)
p(0()) -> 0()
p(s(x)) -> x
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:
{ cond1(true(), x, y) -> cond2(gr(x, y), x, y)
, cond2(true(), x, y) -> cond1(neq(x, 0()), y, y)
, cond2(false(), x, y) -> cond1(neq(x, 0()), p(x), y)
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
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: cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, 2: cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, 3: cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, 4: gr^#(0(), x) -> c_3()
, 5: gr^#(s(x), 0()) -> c_4()
, 6: gr^#(s(x), s(y)) -> c_5(gr^#(x, y))
, 7: neq^#(0(), 0()) -> c_6()
, 8: neq^#(0(), s(x)) -> c_7()
, 9: neq^#(s(x), 0()) -> c_8()
, 10: neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(1)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^2)) ]
|
|->{7} [ YES(?,O(n^2)) ]
|
|->{8} [ YES(?,O(n^2)) ]
|
`->{9} [ YES(?,O(n^3)) ]
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^3)) ]
->{1,3,2} [ NA ]
Sub-problems:
-------------
* Path {1,3,2}: NA
----------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0]
[3 0 0] [3 0 0] [0]
[3 0 0] [3 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(n^2))
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: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {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]
gr^#(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_5(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {6}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_3()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
gr^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_3() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {6}->{5}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
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: {gr^#(s(x), 0()) -> c_4()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
gr^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_4() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {10}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0]
[3 0 0] [3 0 0] [0]
[3 0 0] [3 0 0] [0]
c_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(n^2))
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: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_9) = {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]
neq^#(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_9(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {10}->{7}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {neq^#(0(), 0()) -> c_6()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 3 2] x1 + [0]
[0 1 1] [0]
[0 0 0] [0]
neq^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
[0 4 0] [0 0 0] [2]
[4 0 0] [0 0 0] [4]
c_6() = [1]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 2 0] [0]
* Path {10}->{8}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {neq^#(0(), s(x)) -> c_7()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
neq^#(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_7() = [1]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {10}->{9}: YES(?,O(n^3))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(s(x), 0()) -> c_8()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
neq^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_8() = [1]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
p^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_10() = [0]
[1]
[1]
* Path {12}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
p^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_11() = [0]
[1]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, 2: cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, 3: cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, 4: gr^#(0(), x) -> c_3()
, 5: gr^#(s(x), 0()) -> c_4()
, 6: gr^#(s(x), s(y)) -> c_5(gr^#(x, y))
, 7: neq^#(0(), 0()) -> c_6()
, 8: neq^#(0(), s(x)) -> c_7()
, 9: neq^#(s(x), 0()) -> c_8()
, 10: neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(1)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^2)) ]
|
|->{7} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^2)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: 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:
{ cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {6}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[3 0] [3 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
gr^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_5(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {6}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
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: {gr^#(0(), x) -> c_3()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
gr^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_3() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_4()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
gr^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_4() = [1]
[0]
c_5(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {10}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 1] x1 + [0 1] x2 + [0]
[3 0] [3 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
neq^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_9(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {10}->{7}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {neq^#(0(), 0()) -> c_6()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
neq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[2 0] [0 0] [4]
c_6() = [1]
[0]
c_9(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {10}->{8}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(0(), s(x)) -> c_7()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
neq^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_7() = [1]
[0]
c_9(x1) = [1 0] x1 + [7]
[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 TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {neq^#(s(x), 0()) -> c_8()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
neq^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_8() = [1]
[0]
c_9(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_10() = [0]
[1]
* Path {12}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [2]
[0 0] [2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_11() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, 2: cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, 3: cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, 4: gr^#(0(), x) -> c_3()
, 5: gr^#(s(x), 0()) -> c_4()
, 6: gr^#(s(x), s(y)) -> c_5(gr^#(x, y))
, 7: neq^#(0(), 0()) -> c_6()
, 8: neq^#(0(), s(x)) -> c_7()
, 9: neq^#(s(x), 0()) -> c_8()
, 10: neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(1)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
|->{7} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: 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:
{ cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {gr^#(0(), x) -> c_3()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_3() = [1]
c_5(x1) = [1] x1 + [2]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {gr^#(s(x), 0()) -> c_4()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] 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 TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{7}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(0(), 0()) -> c_6()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
neq^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_6() = [1]
c_9(x1) = [1] x1 + [3]
* Path {10}->{8}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(0(), s(x)) -> c_7()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7() = [1]
c_9(x1) = [1] x1 + [7]
* Path {10}->{9}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(s(x), 0()) -> c_8()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_8() = [1]
c_9(x1) = [1] x1 + [7]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_10() = [1]
* Path {12}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [7]
p^#(x1) = [1] x1 + [7]
c_11() = [1]
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:
{ cond1(true(), x, y) -> cond2(gr(x, y), x, y)
, cond2(true(), x, y) -> cond1(neq(x, 0()), y, y)
, cond2(false(), x, y) -> cond1(neq(x, 0()), p(x), y)
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
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: cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, 2: cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, 3: cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, 4: gr^#(0(), x) -> c_3()
, 5: gr^#(s(x), 0()) -> c_4()
, 6: gr^#(s(x), s(y)) -> c_5(gr^#(x, y))
, 7: neq^#(0(), 0()) -> c_6()
, 8: neq^#(0(), s(x)) -> c_7()
, 9: neq^#(s(x), 0()) -> c_8()
, 10: neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^3)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^2)) ]
|
|->{7} [ YES(?,O(n^2)) ]
|
|->{8} [ YES(?,O(n^2)) ]
|
`->{9} [ YES(?,O(n^3)) ]
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^3)) ]
->{1,3,2} [ NA ]
Sub-problems:
-------------
* Path {1,3,2}: NA
----------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0]
[3 0 0] [3 0 0] [0]
[3 0 0] [3 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^2))
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: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {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]
gr^#(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_5(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {6}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_3()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
gr^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_3() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {6}->{5}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {gr^#(s(x), 0()) -> c_4()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
gr^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_4() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {10}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0]
[3 0 0] [3 0 0] [0]
[3 0 0] [3 0 0] [0]
c_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^2))
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: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_9) = {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]
neq^#(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_9(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {10}->{7}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {neq^#(0(), 0()) -> c_6()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 3 2] x1 + [0]
[0 1 1] [0]
[0 0 0] [0]
neq^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
[0 4 0] [0 0 0] [2]
[4 0 0] [0 0 0] [4]
c_6() = [1]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 2 0] [0]
* Path {10}->{8}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {neq^#(0(), s(x)) -> c_7()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
neq^#(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_7() = [1]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {10}->{9}: YES(?,O(n^3))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(s(x), 0()) -> c_8()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
neq^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_8() = [1]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
p^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_10() = [0]
[1]
[1]
* Path {12}: YES(?,O(n^3))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(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]
true() = [0]
[0]
[0]
cond2(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]
gr(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]
neq(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 3 3] x1 + [0]
[0 1 1] [0]
[0 0 1] [0]
cond1^#(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_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
neq^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^3))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
p^#(x1) = [2 2 2] x1 + [3]
[2 2 0] [7]
[2 2 2] [3]
c_11(x1) = [2 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, 2: cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, 3: cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, 4: gr^#(0(), x) -> c_3()
, 5: gr^#(s(x), 0()) -> c_4()
, 6: gr^#(s(x), s(y)) -> c_5(gr^#(x, y))
, 7: neq^#(0(), 0()) -> c_6()
, 8: neq^#(0(), s(x)) -> c_7()
, 9: neq^#(s(x), 0()) -> c_8()
, 10: neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^2)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^2)) ]
|
|->{7} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^2)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^2)) ]
|
|->{4} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: 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:
{ cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {6}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 1] x1 + [0 1] x2 + [0]
[3 0] [3 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
gr^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_5(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {6}->{4}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {gr^#(0(), x) -> c_3()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
gr^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_3() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_4()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
gr^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_4() = [1]
[0]
c_5(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {10}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 1] x1 + [0 1] x2 + [0]
[3 0] [3 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
neq^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_9(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {10}->{7}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {neq^#(0(), 0()) -> c_6()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
neq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[2 0] [0 0] [4]
c_6() = [1]
[0]
c_9(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {10}->{8}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(0(), s(x)) -> c_7()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
neq^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_7() = [1]
[0]
c_9(x1) = [1 0] x1 + [7]
[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 TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {neq^#(s(x), 0()) -> c_8()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
neq^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_8() = [1]
[0]
c_9(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_10() = [0]
[1]
* Path {12}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 3] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [1 3] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 0] [0]
p^#(x1) = [2 0] x1 + [7]
[2 0] [7]
c_11(x1) = [1 0] x1 + [0]
[0 0] [1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, 2: cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, 3: cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, 4: gr^#(0(), x) -> c_3()
, 5: gr^#(s(x), 0()) -> c_4()
, 6: gr^#(s(x), s(y)) -> c_5(gr^#(x, y))
, 7: neq^#(0(), 0()) -> c_6()
, 8: neq^#(0(), s(x)) -> c_7()
, 9: neq^#(s(x), 0()) -> c_8()
, 10: neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^1)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
|->{7} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(n^1)) ]
|
|->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: 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:
{ cond1^#(true(), x, y) -> c_0(cond2^#(gr(x, y), x, y))
, cond2^#(false(), x, y) -> c_2(cond1^#(neq(x, 0()), p(x), y))
, cond2^#(true(), x, y) -> c_1(cond1^#(neq(x, 0()), y, y))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y)) -> neq(x, y)
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{4}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {gr^#(0(), x) -> c_3()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_3() = [1]
c_5(x1) = [1] x1 + [2]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {gr^#(s(x), 0()) -> c_4()}
Weak Rules: {gr^#(s(x), s(y)) -> c_5(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] 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 TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{7}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(0(), 0()) -> c_6()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
neq^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_6() = [1]
c_9(x1) = [1] x1 + [3]
* Path {10}->{8}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(0(), s(x)) -> c_7()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7() = [1]
c_9(x1) = [1] x1 + [7]
* Path {10}->{9}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
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: {neq^#(s(x), 0()) -> c_8()}
Weak Rules: {neq^#(s(x), s(y)) -> c_9(neq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_8() = [1]
c_9(x1) = [1] x1 + [7]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_10() = [1]
* Path {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(false) = {},
Uargs(p) = {}, Uargs(s) = {}, Uargs(cond1^#) = {}, Uargs(c_0) = {},
Uargs(cond2^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
Uargs(gr^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(c_5) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(p^#) = {},
Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [3] x1 + [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
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: {p^#(s(x)) -> c_11(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [5]
p^#(x1) = [3] x1 + [0]
c_11(x1) = [3] x1 + [0]
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.