Tool CaT
stdout:
MAYBE
Problem:
cond1(true(),x) -> cond2(even(x),x)
cond2(true(),x) -> cond1(neq(x,0()),div2(x))
cond2(false(),x) -> cond1(neq(x,0()),p(x))
neq(0(),0()) -> false()
neq(0(),s(x)) -> true()
neq(s(x),0()) -> true()
neq(s(x),s(y())) -> neq(x,y())
even(0()) -> true()
even(s(0())) -> false()
even(s(s(x))) -> even(x)
div2(0()) -> 0()
div2(s(0())) -> 0()
div2(s(s(x))) -> s(div2(x))
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) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, 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) -> c_0(cond2^#(even(x), x))
, 2: cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, 3: cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, 4: neq^#(0(), 0()) -> c_3()
, 5: neq^#(0(), s(x)) -> c_4()
, 6: neq^#(s(x), 0()) -> c_5()
, 7: neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))
, 8: even^#(0()) -> c_7()
, 9: even^#(s(0())) -> c_8()
, 10: even^#(s(s(x))) -> c_9(even^#(x))
, 11: div2^#(0()) -> c_10()
, 12: div2^#(s(0())) -> c_11()
, 13: div2^#(s(s(x))) -> c_12(div2^#(x))
, 14: p^#(0()) -> c_13()
, 15: p^#(s(x)) -> c_14()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ YES(?,O(1)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(n^2)) ]
|
|->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(n^2)) ]
->{10} [ YES(?,O(n^2)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^2)) ]
->{7} [ YES(?,O(n^2)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ NA ]
Sub-problems:
-------------
* Path {1,3,2}: NA
----------------
The usable rules for this path are:
{ neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, 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 {4}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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: {neq^#(0(), 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(neq^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[3]
neq^#(x1, x2) = [2 0 2] x1 + [1 0 0] x2 + [3]
[0 0 0] [0 2 2] [3]
[2 0 0] [0 2 0] [7]
c_3() = [0]
[1]
[1]
* Path {5}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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: {neq^#(0(), s(x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
neq^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [3]
[0 0 2] [2 0 0] [7]
[0 0 0] [0 2 2] [7]
c_4() = [0]
[1]
[1]
* Path {6}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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: {neq^#(s(x), 0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
neq^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [3]
[0 0 2] [2 0 0] [7]
[0 0 0] [0 2 2] [7]
c_5() = [0]
[1]
[1]
* Path {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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
neq^#(x1, x2) = [0 1 0] x1 + [0 0 0] x2 + [0]
[3 0 0] [0 0 0] [0]
[3 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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_6(neq^#(x, y()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(y) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [0]
[0 1 0] [7]
[0 0 0] [0]
y() = [0]
[1]
[0]
neq^#(x1, x2) = [0 0 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [4 0 0] x1 + [1]
[0 0 0] [0]
[0 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[3 0 0] [0]
[3 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [2 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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: {even^#(s(s(x))) -> c_9(even^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
even^#(x1) = [2 0 2] x1 + [0]
[4 0 2] [0]
[0 0 0] [2]
c_9(x1) = [1 0 2] x1 + [3]
[2 0 0] [0]
[0 0 0] [2]
* 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {even^#(0()) -> c_7()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [2]
[0 0 0] [2]
even^#(x1) = [2 0 0] x1 + [0]
[0 0 2] [4]
[1 2 2] [0]
c_7() = [1]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [6]
* Path {10}->{9}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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: {even^#(s(0())) -> c_8()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 2 0] x1 + [2]
[0 1 2] [0]
[0 0 0] [0]
even^#(x1) = [2 3 0] x1 + [0]
[0 0 0] [0]
[0 5 0] [0]
c_8() = [1]
[0]
[0]
c_9(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {13}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[3 0 0] [0]
[3 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(x1) = [2 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_13() = [0]
[0]
[0]
c_14() = [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: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
div2^#(x1) = [2 0 2] x1 + [0]
[4 0 2] [0]
[0 0 0] [2]
c_12(x1) = [1 0 2] x1 + [3]
[2 0 0] [0]
[0 0 0] [2]
* Path {13}->{11}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {div2^#(0()) -> c_10()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [2]
[0 0 0] [2]
div2^#(x1) = [2 0 0] x1 + [0]
[0 0 2] [4]
[1 2 2] [0]
c_10() = [1]
[0]
[0]
c_12(x1) = [1 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [6]
* Path {13}->{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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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: {div2^#(s(0())) -> c_11()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 2 0] x1 + [2]
[0 1 2] [0]
[0 0 0] [0]
div2^#(x1) = [2 3 0] x1 + [0]
[0 0 0] [0]
[0 5 0] [0]
c_11() = [1]
[0]
[0]
c_12(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {14}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_13) = {}
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_13() = [0]
[1]
[1]
* Path {15}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14() = [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_14()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_14) = {}
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_14() = [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) -> c_0(cond2^#(even(x), x))
, 2: cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, 3: cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, 4: neq^#(0(), 0()) -> c_3()
, 5: neq^#(0(), s(x)) -> c_4()
, 6: neq^#(s(x), 0()) -> c_5()
, 7: neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))
, 8: even^#(0()) -> c_7()
, 9: even^#(s(0())) -> c_8()
, 10: even^#(s(s(x))) -> c_9(even^#(x))
, 11: div2^#(0()) -> c_10()
, 12: div2^#(s(0())) -> c_11()
, 13: div2^#(s(s(x))) -> c_12(div2^#(x))
, 14: p^#(0()) -> c_13()
, 15: p^#(s(x)) -> c_14()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ YES(?,O(1)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(n^1)) ]
|
|->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(n^2)) ]
->{10} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^2)) ]
->{7} [ YES(?,O(n^2)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, 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) -> c_0(cond2^#(even(x), x))
, cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {neq^#(0(), 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(neq^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
neq^#(x1, x2) = [2 2] x1 + [0 0] x2 + [7]
[2 2] [0 2] [3]
c_3() = [0]
[1]
* Path {5}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {neq^#(0(), s(x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [0 0] x1 + [2]
[0 0] [0]
neq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [3]
c_4() = [0]
[1]
* Path {6}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {neq^#(s(x), 0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [0 0] x1 + [2]
[0 0] [2]
neq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [3]
c_5() = [0]
[1]
* Path {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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [2 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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_6(neq^#(x, y()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(y) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 1] [1]
y() = [2]
[0]
neq^#(x1, x2) = [2 1] x1 + [0 0] x2 + [0]
[0 1] [2 0] [0]
c_6(x1) = [1 1] x1 + [0]
[0 0] [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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[3 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [2 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {even^#(s(s(x))) -> c_9(even^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
even^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_9(x1) = [1 0] x1 + [5]
[2 0] [7]
* 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {even^#(0()) -> c_7()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [0]
even^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_7() = [1]
[0]
c_9(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {10}->{9}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {even^#(s(0())) -> c_8()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [1]
[0]
s(x1) = [1 1] x1 + [2]
[0 1] [1]
even^#(x1) = [2 1] x1 + [1]
[0 0] [7]
c_8() = [1]
[1]
c_9(x1) = [1 1] x1 + [3]
[0 0] [3]
* Path {13}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[3 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [2 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
div2^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_12(x1) = [1 0] x1 + [5]
[2 0] [7]
* Path {13}->{11}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {div2^#(0()) -> c_10()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [0]
div2^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_10() = [1]
[0]
c_12(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {13}->{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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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: {div2^#(s(0())) -> c_11()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [1]
[0]
s(x1) = [1 1] x1 + [2]
[0 1] [1]
div2^#(x1) = [2 1] x1 + [1]
[0 0] [7]
c_11() = [1]
[1]
c_12(x1) = [1 1] x1 + [3]
[0 0] [3]
* Path {14}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_13() = [0]
[1]
* Path {15}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14() = [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_14()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_14) = {}
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_14() = [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) -> c_0(cond2^#(even(x), x))
, 2: cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, 3: cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, 4: neq^#(0(), 0()) -> c_3()
, 5: neq^#(0(), s(x)) -> c_4()
, 6: neq^#(s(x), 0()) -> c_5()
, 7: neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))
, 8: even^#(0()) -> c_7()
, 9: even^#(s(0())) -> c_8()
, 10: even^#(s(s(x))) -> c_9(even^#(x))
, 11: div2^#(0()) -> c_10()
, 12: div2^#(s(0())) -> c_11()
, 13: div2^#(s(s(x))) -> c_12(div2^#(x))
, 14: p^#(0()) -> c_13()
, 15: p^#(s(x)) -> c_14()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ YES(?,O(1)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(n^1)) ]
|
|->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(1)) ]
->{7} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, 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) -> c_0(cond2^#(even(x), x))
, cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {neq^#(0(), 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(neq^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_3() = [0]
* Path {5}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {neq^#(0(), s(x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [0] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_4() = [0]
* Path {6}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {neq^#(s(x), 0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [0] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_5() = [0]
* Path {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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(y) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [2]
y() = [0]
neq^#(x1, x2) = [0] x1 + [2] x2 + [0]
c_6(x1) = [2] x1 + [3]
* 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [3] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {even^#(s(s(x))) -> c_9(even^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
even^#(x1) = [2] x1 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{8}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {even^#(0()) -> c_7()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
even^#(x1) = [2] x1 + [4]
c_7() = [1]
c_9(x1) = [1] x1 + [0]
* Path {10}->{9}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {even^#(s(0())) -> c_8()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [0]
even^#(x1) = [0] x1 + [1]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
* Path {13}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [3] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [1] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
div2^#(x1) = [2] x1 + [0]
c_12(x1) = [1] x1 + [7]
* Path {13}->{11}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {div2^#(0()) -> c_10()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
div2^#(x1) = [2] x1 + [4]
c_10() = [1]
c_12(x1) = [1] x1 + [0]
* Path {13}->{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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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: {div2^#(s(0())) -> c_11()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [0]
div2^#(x1) = [0] x1 + [1]
c_11() = [0]
c_12(x1) = [1] x1 + [0]
* Path {14}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_13() = [1]
* Path {15}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14() = [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_14()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [7]
p^#(x1) = [1] x1 + [7]
c_14() = [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) -> cond2(even(x), x)
, cond2(true(), x) -> cond1(neq(x, 0()), div2(x))
, cond2(false(), x) -> cond1(neq(x, 0()), p(x))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, 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) -> c_0(cond2^#(even(x), x))
, 2: cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, 3: cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, 4: neq^#(0(), 0()) -> c_3()
, 5: neq^#(0(), s(x)) -> c_4()
, 6: neq^#(s(x), 0()) -> c_5()
, 7: neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))
, 8: even^#(0()) -> c_7()
, 9: even^#(s(0())) -> c_8()
, 10: even^#(s(s(x))) -> c_9(even^#(x))
, 11: div2^#(0()) -> c_10()
, 12: div2^#(s(0())) -> c_11()
, 13: div2^#(s(s(x))) -> c_12(div2^#(x))
, 14: p^#(0()) -> c_13()
, 15: p^#(s(x)) -> c_14(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ YES(?,O(n^3)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(n^2)) ]
|
|->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(n^2)) ]
->{10} [ YES(?,O(n^2)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^2)) ]
->{7} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ NA ]
Sub-problems:
-------------
* Path {1,3,2}: NA
----------------
The usable rules for this path are:
{ neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, 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 {4}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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: {neq^#(0(), 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(neq^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[3]
neq^#(x1, x2) = [2 0 2] x1 + [1 0 0] x2 + [3]
[0 0 0] [0 2 2] [3]
[2 0 0] [0 2 0] [7]
c_3() = [0]
[1]
[1]
* Path {5}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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: {neq^#(0(), s(x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
neq^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [3]
[0 0 2] [2 0 0] [7]
[0 0 0] [0 2 2] [7]
c_4() = [0]
[1]
[1]
* Path {6}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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: {neq^#(s(x), 0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [0 0 0] x1 + [2]
[0 0 0] [2]
[0 0 0] [2]
neq^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [3]
[0 0 2] [2 0 0] [7]
[0 0 0] [0 2 2] [7]
c_5() = [0]
[1]
[1]
* Path {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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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 0 0] [0]
[0 0 0] [0]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
neq^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 0 0] [0 0 0] [0]
[3 0 0] [0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [2 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[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 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(y) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0 0] x1 + [0]
[0 1 0] [7]
[0 0 0] [0]
y() = [0]
[1]
[0]
neq^#(x1, x2) = [0 0 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_6(x1) = [4 0 0] x1 + [1]
[0 0 0] [0]
[0 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[3 0 0] [0]
[3 0 0] [0]
c_7() = [0]
[0]
[0]
c_8() = [0]
[0]
[0]
c_9(x1) = [2 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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: {even^#(s(s(x))) -> c_9(even^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
even^#(x1) = [2 0 2] x1 + [0]
[4 0 2] [0]
[0 0 0] [2]
c_9(x1) = [1 0 2] x1 + [3]
[2 0 0] [0]
[0 0 0] [2]
* 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {even^#(0()) -> c_7()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [2]
[0 0 0] [2]
even^#(x1) = [2 0 0] x1 + [0]
[0 0 2] [4]
[1 2 2] [0]
c_7() = [1]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [6]
* Path {10}->{9}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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: {even^#(s(0())) -> c_8()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 2 0] x1 + [2]
[0 1 2] [0]
[0 0 0] [0]
even^#(x1) = [2 3 0] x1 + [0]
[0 0 0] [0]
[0 5 0] [0]
c_8() = [1]
[0]
[0]
c_9(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {13}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[3 0 0] [0]
[3 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(x1) = [2 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_13() = [0]
[0]
[0]
c_14(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: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
div2^#(x1) = [2 0 2] x1 + [0]
[4 0 2] [0]
[0 0 0] [2]
c_12(x1) = [1 0 2] x1 + [3]
[2 0 0] [0]
[0 0 0] [2]
* Path {13}->{11}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {div2^#(0()) -> c_10()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [2]
[0 0 0] [2]
div2^#(x1) = [2 0 0] x1 + [0]
[0 0 2] [4]
[1 2 2] [0]
c_10() = [1]
[0]
[0]
c_12(x1) = [1 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [6]
* Path {13}->{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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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: {div2^#(s(0())) -> c_11()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 2 0] x1 + [2]
[0 1 2] [0]
[0 0 0] [0]
div2^#(x1) = [2 3 0] x1 + [0]
[0 0 0] [0]
[0 5 0] [0]
c_11() = [1]
[0]
[0]
c_12(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {14}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(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_13() = [0]
[0]
[0]
c_14(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_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_13) = {}
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_13() = [0]
[1]
[1]
* Path {15}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
cond2(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]
even(x1) = [0 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]
0() = [0]
[0]
[0]
div2(x1) = [0 0 0] x1 + [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 2] [0]
[0 0 1] [0]
y() = [0]
[0]
[0]
cond1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(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_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]
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_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
c_5() = [0]
[0]
[0]
c_6(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
even^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 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]
div2^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11() = [0]
[0]
[0]
c_12(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
p^#(x1) = [1 3 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_13() = [0]
[0]
[0]
c_14(x1) = [1 0 0] 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_14(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_14) = {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_14(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) -> c_0(cond2^#(even(x), x))
, 2: cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, 3: cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, 4: neq^#(0(), 0()) -> c_3()
, 5: neq^#(0(), s(x)) -> c_4()
, 6: neq^#(s(x), 0()) -> c_5()
, 7: neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))
, 8: even^#(0()) -> c_7()
, 9: even^#(s(0())) -> c_8()
, 10: even^#(s(s(x))) -> c_9(even^#(x))
, 11: div2^#(0()) -> c_10()
, 12: div2^#(s(0())) -> c_11()
, 13: div2^#(s(s(x))) -> c_12(div2^#(x))
, 14: p^#(0()) -> c_13()
, 15: p^#(s(x)) -> c_14(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ YES(?,O(n^2)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(n^1)) ]
|
|->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(n^2)) ]
->{10} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^2)) ]
->{7} [ YES(?,O(n^2)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, 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) -> c_0(cond2^#(even(x), x))
, cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {neq^#(0(), 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(neq^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
neq^#(x1, x2) = [2 2] x1 + [0 0] x2 + [7]
[2 2] [0 2] [3]
c_3() = [0]
[1]
* Path {5}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {neq^#(0(), s(x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [0 0] x1 + [2]
[0 0] [0]
neq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [3]
c_4() = [0]
[1]
* Path {6}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {neq^#(s(x), 0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [0 0] x1 + [2]
[0 0] [2]
neq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [3]
c_5() = [0]
[1]
* Path {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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [2 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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_6(neq^#(x, y()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(y) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 1] [1]
y() = [2]
[0]
neq^#(x1, x2) = [2 1] x1 + [0 0] x2 + [0]
[0 1] [2 0] [0]
c_6(x1) = [1 1] x1 + [0]
[0 0] [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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[3 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [2 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {even^#(s(s(x))) -> c_9(even^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
even^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_9(x1) = [1 0] x1 + [5]
[2 0] [7]
* 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {even^#(0()) -> c_7()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [0]
even^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_7() = [1]
[0]
c_9(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {10}->{9}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {even^#(s(0())) -> c_8()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [1]
[0]
s(x1) = [1 1] x1 + [2]
[0 1] [1]
even^#(x1) = [2 1] x1 + [1]
[0 0] [7]
c_8() = [1]
[1]
c_9(x1) = [1 1] x1 + [3]
[0 0] [3]
* Path {13}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[3 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [2 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
div2^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_12(x1) = [1 0] x1 + [5]
[2 0] [7]
* Path {13}->{11}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {div2^#(0()) -> c_10()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [0]
div2^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_10() = [1]
[0]
c_12(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {13}->{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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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: {div2^#(s(0())) -> c_11()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [1]
[0]
s(x1) = [1 1] x1 + [2]
[0 1] [1]
div2^#(x1) = [2 1] x1 + [1]
[0 0] [7]
c_11() = [1]
[1]
c_12(x1) = [1 1] x1 + [3]
[0 0] [3]
* Path {14}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_13() = [0]
[1]
* Path {15}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
even(x1) = [0 0] x1 + [0]
[0 0] [0]
neq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
div2(x1) = [0 0] x1 + [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]
y() = [0]
[0]
cond1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
neq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [3 0] x1 + [0]
[0 0] [0]
even^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
div2^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
c_12(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [1 3] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(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_14(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_14) = {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_14(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) -> c_0(cond2^#(even(x), x))
, 2: cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, 3: cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, 4: neq^#(0(), 0()) -> c_3()
, 5: neq^#(0(), s(x)) -> c_4()
, 6: neq^#(s(x), 0()) -> c_5()
, 7: neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))
, 8: even^#(0()) -> c_7()
, 9: even^#(s(0())) -> c_8()
, 10: even^#(s(s(x))) -> c_9(even^#(x))
, 11: div2^#(0()) -> c_10()
, 12: div2^#(s(0())) -> c_11()
, 13: div2^#(s(s(x))) -> c_12(div2^#(x))
, 14: p^#(0()) -> c_13()
, 15: p^#(s(x)) -> c_14(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ YES(?,O(n^1)) ]
->{14} [ YES(?,O(1)) ]
->{13} [ YES(?,O(n^1)) ]
|
|->{11} [ YES(?,O(n^1)) ]
|
`->{12} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
|->{8} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(1)) ]
->{7} [ YES(?,O(n^1)) ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{1,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,3,2}: MAYBE
-------------------
The usable rules for this path are:
{ neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, 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) -> c_0(cond2^#(even(x), x))
, cond2^#(false(), x) -> c_2(cond1^#(neq(x, 0()), p(x)))
, cond2^#(true(), x) -> c_1(cond1^#(neq(x, 0()), div2(x)))
, neq(0(), 0()) -> false()
, neq(0(), s(x)) -> true()
, neq(s(x), 0()) -> true()
, neq(s(x), s(y())) -> neq(x, y())
, even(0()) -> true()
, even(s(0())) -> false()
, even(s(s(x))) -> even(x)
, div2(0()) -> 0()
, div2(s(0())) -> 0()
, div2(s(s(x))) -> s(div2(x))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {4}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {neq^#(0(), 0()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(neq^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_3() = [0]
* Path {5}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {neq^#(0(), s(x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [0] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_4() = [0]
* Path {6}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {neq^#(s(x), 0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(neq^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [0] x1 + [2]
neq^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_5() = [0]
* Path {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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {neq^#(s(x), s(y())) -> c_6(neq^#(x, y()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(y) = {}, Uargs(neq^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [2]
y() = [0]
neq^#(x1, x2) = [0] x1 + [2] x2 + [0]
c_6(x1) = [2] x1 + [3]
* 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {even^#(s(s(x))) -> c_9(even^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
even^#(x1) = [2] x1 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{8}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {even^#(0()) -> c_7()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_7) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
even^#(x1) = [2] x1 + [4]
c_7() = [1]
c_9(x1) = [1] x1 + [0]
* Path {10}->{9}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {even^#(s(0())) -> c_8()}
Weak Rules: {even^#(s(s(x))) -> c_9(even^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(even^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [0]
even^#(x1) = [0] x1 + [1]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
* Path {13}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
div2^#(x1) = [2] x1 + [0]
c_12(x1) = [1] x1 + [7]
* Path {13}->{11}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {div2^#(0()) -> c_10()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
div2^#(x1) = [2] x1 + [4]
c_10() = [1]
c_12(x1) = [1] x1 + [0]
* Path {13}->{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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {1}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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: {div2^#(s(0())) -> c_11()}
Weak Rules: {div2^#(s(s(x))) -> c_12(div2^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(div2^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [0]
div2^#(x1) = [0] x1 + [1]
c_11() = [0]
c_12(x1) = [1] x1 + [0]
* Path {14}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(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_13()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_13() = [1]
* Path {15}: 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(even) = {}, Uargs(neq) = {}, Uargs(0) = {}, Uargs(div2) = {},
Uargs(false) = {}, Uargs(p) = {}, Uargs(s) = {}, Uargs(y) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(neq^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
Uargs(even^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(div2^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(p^#) = {},
Uargs(c_13) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
cond2(x1, x2) = [0] x1 + [0] x2 + [0]
even(x1) = [0] x1 + [0]
neq(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
div2(x1) = [0] x1 + [0]
false() = [0]
p(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
y() = [0]
cond1^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
neq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [3] x1 + [0]
even^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
div2^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
c_12(x1) = [3] x1 + [0]
p^#(x1) = [3] x1 + [0]
c_13() = [0]
c_14(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_14(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [5]
p^#(x1) = [3] x1 + [0]
c_14(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.