Tool CaT
stdout:
MAYBE
Problem:
diff(x,y) -> cond1(equal(x,y),x,y)
cond1(true(),x,y) -> 0()
cond1(false(),x,y) -> cond2(gt(x,y),x,y)
cond2(true(),x,y) -> s(diff(x,s(y)))
cond2(false(),x,y) -> s(diff(s(x),y))
gt(0(),v) -> false()
gt(s(u),0()) -> true()
gt(s(u),s(v)) -> gt(u,v)
equal(0(),0()) -> true()
equal(s(x),0()) -> false()
equal(0(),s(y)) -> false()
equal(s(x),s(y)) -> equal(x,y)
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:
{ diff(x, y) -> cond1(equal(x, y), x, y)
, cond1(true(), x, y) -> 0()
, cond1(false(), x, y) -> cond2(gt(x, y), x, y)
, cond2(true(), x, y) -> s(diff(x, s(y)))
, cond2(false(), x, y) -> s(diff(s(x), y))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
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: diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, 2: cond1^#(true(), x, y) -> c_1()
, 3: cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, 4: cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, 5: cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, 6: gt^#(0(), v) -> c_5()
, 7: gt^#(s(u), 0()) -> c_6()
, 8: gt^#(s(u), s(v)) -> c_7(gt^#(u, v))
, 9: equal^#(0(), 0()) -> c_8()
, 10: equal^#(s(x), 0()) -> c_9()
, 11: equal^#(0(), s(y)) -> c_10()
, 12: equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^1)) ]
|
|->{9} [ YES(?,O(n^2)) ]
|
|->{10} [ YES(?,O(n^3)) ]
|
`->{11} [ YES(?,O(n^2)) ]
->{8} [ YES(?,O(n^1)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{1,5,3,4} [ NA ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,5,3,4}: NA
------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {1,5,3,4}->{2}: NA
-----------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {1}, Uargs(cond1^#) = {1},
Uargs(c_2) = {1}, Uargs(cond2^#) = {1}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(x1, x2) = [0 0 3] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 2] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [2]
diff^#(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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
cond1^#(x1, x2, x3) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
cond2^#(x1, x2, x3) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
gt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_7(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {8}->{6}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_5()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
gt^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 0] [0 4 0] [0]
c_5() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {8}->{7}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_6()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 2] [2]
[0 0 0] [2]
gt^#(x1, x2) = [0 2 2] x1 + [0 0 0] x2 + [0]
[0 0 0] [4 0 0] [4]
[0 0 0] [0 0 0] [2]
c_6() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
* Path {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
equal^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_11(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {12}->{9}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), 0()) -> c_8()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 3 2] x1 + [0]
[0 1 1] [0]
[0 0 0] [0]
equal^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
[0 4 0] [0 0 0] [2]
[4 0 0] [0 0 0] [4]
c_8() = [1]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 2 0] [0]
* Path {12}->{10}: YES(?,O(n^3))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), 0()) -> c_9()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
equal^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_9() = [1]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {12}->{11}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), s(y)) -> c_10()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
equal^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_10() = [1]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, 2: cond1^#(true(), x, y) -> c_1()
, 3: cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, 4: cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, 5: cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, 6: gt^#(0(), v) -> c_5()
, 7: gt^#(s(u), 0()) -> c_6()
, 8: gt^#(s(u), s(v)) -> c_7(gt^#(u, v))
, 9: equal^#(0(), 0()) -> c_8()
, 10: equal^#(s(x), 0()) -> c_9()
, 11: equal^#(0(), s(y)) -> c_10()
, 12: equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^2)) ]
|
|->{9} [ YES(?,O(n^1)) ]
|
|->{10} [ YES(?,O(n^1)) ]
|
`->{11} [ YES(?,O(n^2)) ]
->{8} [ YES(?,O(n^2)) ]
|
|->{6} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{1,5,3,4} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,5,3,4}: MAYBE
---------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1,5,3,4}->{2}: NA
-----------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {1}, Uargs(cond1^#) = {1},
Uargs(c_2) = {1}, Uargs(cond2^#) = {1}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [3]
true() = [0]
[1]
0() = [0]
[0]
false() = [0]
[1]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 1] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
cond2^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_7(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {8}->{6}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_5()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_5() = [1]
[0]
c_7(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {8}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_6()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [2]
gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_6() = [1]
[0]
c_7(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {12}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
equal^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_11(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {12}->{9}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), 0()) -> c_8()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
equal^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[2 0] [0 0] [4]
c_8() = [1]
[0]
c_11(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {12}->{10}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), 0()) -> c_9()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
equal^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_9() = [1]
[0]
c_11(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {12}->{11}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), s(y)) -> c_10()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
equal^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_10() = [1]
[0]
c_11(x1) = [1 0] x1 + [7]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, 2: cond1^#(true(), x, y) -> c_1()
, 3: cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, 4: cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, 5: cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, 6: gt^#(0(), v) -> c_5()
, 7: gt^#(s(u), 0()) -> c_6()
, 8: gt^#(s(u), s(v)) -> c_7(gt^#(u, v))
, 9: equal^#(0(), 0()) -> c_8()
, 10: equal^#(s(x), 0()) -> c_9()
, 11: equal^#(0(), s(y)) -> c_10()
, 12: equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^1)) ]
|
|->{9} [ YES(?,O(n^1)) ]
|
|->{10} [ YES(?,O(n^1)) ]
|
`->{11} [ YES(?,O(n^1)) ]
->{8} [ YES(?,O(n^1)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{1,5,3,4} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,5,3,4}: MAYBE
---------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1,5,3,4}->{2}: NA
-----------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {1}, Uargs(cond1^#) = {1},
Uargs(c_2) = {1}, Uargs(cond2^#) = {1}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [2] x2 + [1]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [2] x2 + [1]
s(x1) = [1] x1 + [2]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
cond2^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7(x1) = [1] x1 + [7]
* Path {8}->{6}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_5()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_5() = [1]
c_7(x1) = [1] x1 + [5]
* Path {8}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_6()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6() = [1]
c_7(x1) = [1] x1 + [7]
* Path {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_11(x1) = [1] x1 + [7]
* Path {12}->{9}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), 0()) -> c_8()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
equal^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_8() = [1]
c_11(x1) = [1] x1 + [3]
* Path {12}->{10}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), 0()) -> c_9()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_9() = [1]
c_11(x1) = [1] x1 + [7]
* Path {12}->{11}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), s(y)) -> c_10()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_10() = [1]
c_11(x1) = [1] x1 + [7]
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:
{ diff(x, y) -> cond1(equal(x, y), x, y)
, cond1(true(), x, y) -> 0()
, cond1(false(), x, y) -> cond2(gt(x, y), x, y)
, cond2(true(), x, y) -> s(diff(x, s(y)))
, cond2(false(), x, y) -> s(diff(s(x), y))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
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: diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, 2: cond1^#(true(), x, y) -> c_1()
, 3: cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, 4: cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, 5: cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, 6: gt^#(0(), v) -> c_5()
, 7: gt^#(s(u), 0()) -> c_6()
, 8: gt^#(s(u), s(v)) -> c_7(gt^#(u, v))
, 9: equal^#(0(), 0()) -> c_8()
, 10: equal^#(s(x), 0()) -> c_9()
, 11: equal^#(0(), s(y)) -> c_10()
, 12: equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^1)) ]
|
|->{9} [ YES(?,O(n^2)) ]
|
|->{10} [ YES(?,O(n^3)) ]
|
`->{11} [ YES(?,O(n^2)) ]
->{8} [ YES(?,O(n^1)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^2)) ]
->{1,5,3,4} [ NA ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,5,3,4}: NA
------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {1,5,3,4}->{2}: NA
-----------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {1}, Uargs(cond1^#) = {1},
Uargs(c_2) = {1}, Uargs(cond2^#) = {1}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(x1, x2) = [0 0 3] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 2] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [2]
diff^#(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) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
cond1^#(x1, x2, x3) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
cond2^#(x1, x2, x3) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
gt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_7(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {8}->{6}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_5()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
gt^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 0] [0 4 0] [0]
c_5() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {8}->{7}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_6()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 2] [2]
[0 0 0] [2]
gt^#(x1, x2) = [0 2 2] x1 + [0 0 0] x2 + [0]
[0 0 0] [4 0 0] [4]
[0 0 0] [0 0 0] [2]
c_6() = [1]
[0]
[0]
c_7(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
* Path {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
equal^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_11(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {12}->{9}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), 0()) -> c_8()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 3 2] x1 + [0]
[0 1 1] [0]
[0 0 0] [0]
equal^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
[0 4 0] [0 0 0] [2]
[4 0 0] [0 0 0] [4]
c_8() = [1]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 2 0] [0]
* Path {12}->{10}: YES(?,O(n^3))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), 0()) -> c_9()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
equal^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_9() = [1]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {12}->{11}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(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]
cond1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
equal(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]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
cond2(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
diff^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond1^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond2^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_5() = [0]
[0]
[0]
c_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
equal^#(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_8() = [0]
[0]
[0]
c_9() = [0]
[0]
[0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), s(y)) -> c_10()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
equal^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_10() = [1]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, 2: cond1^#(true(), x, y) -> c_1()
, 3: cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, 4: cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, 5: cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, 6: gt^#(0(), v) -> c_5()
, 7: gt^#(s(u), 0()) -> c_6()
, 8: gt^#(s(u), s(v)) -> c_7(gt^#(u, v))
, 9: equal^#(0(), 0()) -> c_8()
, 10: equal^#(s(x), 0()) -> c_9()
, 11: equal^#(0(), s(y)) -> c_10()
, 12: equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^2)) ]
|
|->{9} [ YES(?,O(n^1)) ]
|
|->{10} [ YES(?,O(n^1)) ]
|
`->{11} [ YES(?,O(n^2)) ]
->{8} [ YES(?,O(n^2)) ]
|
|->{6} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{1,5,3,4} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,5,3,4}: MAYBE
---------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1,5,3,4}->{2}: NA
-----------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {1}, Uargs(cond1^#) = {1},
Uargs(c_2) = {1}, Uargs(cond2^#) = {1}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [3]
true() = [0]
[1]
0() = [0]
[0]
false() = [0]
[1]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 1] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
cond2^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_7(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {8}->{6}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_5()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_5() = [1]
[0]
c_7(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {8}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_6()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [2]
gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_6() = [1]
[0]
c_7(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {12}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
equal^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_11(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {12}->{9}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), 0()) -> c_8()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
equal^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[2 0] [0 0] [4]
c_8() = [1]
[0]
c_11(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {12}->{10}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), 0()) -> c_9()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
equal^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_9() = [1]
[0]
c_11(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {12}->{11}: YES(?,O(n^2))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cond1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
equal(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
0() = [0]
[0]
false() = [0]
[0]
cond2(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
diff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), s(y)) -> c_10()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
equal^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_10() = [1]
[0]
c_11(x1) = [1 0] x1 + [7]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, 2: cond1^#(true(), x, y) -> c_1()
, 3: cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, 4: cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, 5: cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, 6: gt^#(0(), v) -> c_5()
, 7: gt^#(s(u), 0()) -> c_6()
, 8: gt^#(s(u), s(v)) -> c_7(gt^#(u, v))
, 9: equal^#(0(), 0()) -> c_8()
, 10: equal^#(s(x), 0()) -> c_9()
, 11: equal^#(0(), s(y)) -> c_10()
, 12: equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^1)) ]
|
|->{9} [ YES(?,O(n^1)) ]
|
|->{10} [ YES(?,O(n^1)) ]
|
`->{11} [ YES(?,O(n^1)) ]
->{8} [ YES(?,O(n^1)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{1,5,3,4} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {1,5,3,4}: MAYBE
---------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ diff^#(x, y) -> c_0(cond1^#(equal(x, y), x, y))
, cond2^#(false(), x, y) -> c_4(diff^#(s(x), y))
, cond1^#(false(), x, y) -> c_2(cond2^#(gt(x, y), x, y))
, cond2^#(true(), x, y) -> c_3(diff^#(x, s(y)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {1,5,3,4}->{2}: NA
-----------------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {1}, Uargs(cond1^#) = {1},
Uargs(c_2) = {1}, Uargs(cond2^#) = {1}, Uargs(c_3) = {1},
Uargs(c_4) = {1}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [2] x2 + [1]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [2] x2 + [1]
s(x1) = [1] x1 + [2]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
cond2^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7(x1) = [1] x1 + [7]
* Path {8}->{6}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_5()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_5() = [1]
c_7(x1) = [1] x1 + [5]
* Path {8}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1},
Uargs(equal^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_6()}
Weak Rules: {gt^#(s(u), s(v)) -> c_7(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6() = [1]
c_7(x1) = [1] x1 + [7]
* Path {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_11(x1) = [1] x1 + [7]
* Path {12}->{9}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), 0()) -> c_8()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
equal^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_8() = [1]
c_11(x1) = [1] x1 + [3]
* Path {12}->{10}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(s(x), 0()) -> c_9()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_9() = [1]
c_11(x1) = [1] x1 + [7]
* Path {12}->{11}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(diff) = {}, Uargs(cond1) = {}, Uargs(equal) = {},
Uargs(cond2) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(diff^#) = {}, Uargs(c_0) = {}, Uargs(cond1^#) = {},
Uargs(c_2) = {}, Uargs(cond2^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(gt^#) = {}, Uargs(c_7) = {},
Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
diff(x1, x2) = [0] x1 + [0] x2 + [0]
cond1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
equal(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
0() = [0]
false() = [0]
cond2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
diff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
cond2^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9() = [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {equal^#(0(), s(y)) -> c_10()}
Weak Rules: {equal^#(s(x), s(y)) -> c_11(equal^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_10() = [1]
c_11(x1) = [1] x1 + [7]
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.