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