Tool CaT
stdout:
MAYBE
Problem:
cond1(true(),x,y,z) -> cond2(gr(x,0()),x,y,z)
cond2(true(),x,y,z) -> cond1(gr(add(x,y),z),p(x),y,z)
cond2(false(),x,y,z) -> cond3(gr(y,0()),x,y,z)
cond3(true(),x,y,z) -> cond1(gr(add(x,y),z),x,p(y),z)
cond3(false(),x,y,z) -> cond1(gr(add(x,y),z),x,y,z)
gr(0(),x) -> false()
gr(s(x),0()) -> true()
gr(s(x),s(y)) -> gr(x,y)
add(0(),x) -> x
add(s(x),y) -> s(add(x,y))
p(0()) -> 0()
p(s(x)) -> x
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ cond1(true(), x, y, z) -> cond2(gr(x, 0()), x, y, z)
, cond2(true(), x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
, cond2(false(), x, y, z) -> cond3(gr(y, 0()), x, y, z)
, cond3(true(), x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
, cond3(false(), x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond1^#(true(), x, y, z) -> c_0(cond2^#(gr(x, 0()), x, y, z))
, 2: cond2^#(true(), x, y, z) ->
c_1(cond1^#(gr(add(x, y), z), p(x), y, z))
, 3: cond2^#(false(), x, y, z) -> c_2(cond3^#(gr(y, 0()), x, y, z))
, 4: cond3^#(true(), x, y, z) ->
c_3(cond1^#(gr(add(x, y), z), x, p(y), z))
, 5: cond3^#(false(), x, y, z) ->
c_4(cond1^#(gr(add(x, y), z), x, y, z))
, 6: gr^#(0(), x) -> c_5()
, 7: gr^#(s(x), 0()) -> c_6()
, 8: gr^#(s(x), s(y)) -> c_7(gr^#(x, y))
, 9: add^#(0(), x) -> c_8()
, 10: add^#(s(x), y) -> c_9(add^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(1)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^2)) ]
->{8} [ YES(?,O(n^2)) ]
|
|->{6} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{1,5,3,4,2} [ NA ]
Sub-problems:
-------------
* Path {1,5,3,4,2}: NA
--------------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 1] x1 + [0 1] x2 + [0]
[3 0] [3 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
gr^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_7(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {8}->{6}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_5()}
Weak Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
gr^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_5() = [1]
[0]
c_7(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {8}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_6()}
Weak Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
gr^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_6() = [1]
[0]
c_7(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {10}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 1] x1 + [3 3] x2 + [0]
[3 0] [3 3] [0]
c_8() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_9(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_9(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {10}->{9}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_8()}
Weak Rules: {add^#(s(x), y) -> c_9(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [4]
[0 1] [0]
add^#(x1, x2) = [1 3] x1 + [0 0] x2 + [4]
[2 2] [4 4] [0]
c_8() = [1]
[0]
c_9(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_10() = [0]
[1]
* Path {12}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11() = [0]
[0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [2]
[0 0] [2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_11() = [0]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond1^#(true(), x, y, z) -> c_0(cond2^#(gr(x, 0()), x, y, z))
, 2: cond2^#(true(), x, y, z) ->
c_1(cond1^#(gr(add(x, y), z), p(x), y, z))
, 3: cond2^#(false(), x, y, z) -> c_2(cond3^#(gr(y, 0()), x, y, z))
, 4: cond3^#(true(), x, y, z) ->
c_3(cond1^#(gr(add(x, y), z), x, p(y), z))
, 5: cond3^#(false(), x, y, z) ->
c_4(cond1^#(gr(add(x, y), z), x, y, z))
, 6: gr^#(0(), x) -> c_5()
, 7: gr^#(s(x), 0()) -> c_6()
, 8: gr^#(s(x), s(y)) -> c_7(gr^#(x, y))
, 9: add^#(0(), x) -> c_8()
, 10: add^#(s(x), y) -> c_9(add^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(1)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{8} [ YES(?,O(n^1)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{1,5,3,4,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,5,3,4,2}: MAYBE
-----------------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ cond1^#(true(), x, y, z) -> c_0(cond2^#(gr(x, 0()), x, y, z))
, cond3^#(false(), x, y, z) ->
c_4(cond1^#(gr(add(x, y), z), x, y, z))
, cond2^#(false(), x, y, z) -> c_2(cond3^#(gr(y, 0()), x, y, z))
, cond3^#(true(), x, y, z) ->
c_3(cond1^#(gr(add(x, y), z), x, p(y), z))
, cond2^#(true(), x, y, z) ->
c_1(cond1^#(gr(add(x, y), z), p(x), y, z))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7(x1) = [1] x1 + [7]
* Path {8}->{6}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_5()}
Weak Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_5() = [1]
c_7(x1) = [1] x1 + [2]
* Path {8}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_6()}
Weak Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6() = [1]
c_7(x1) = [1] x1 + [7]
* Path {10}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_9(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
add^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{9}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_8()}
Weak Rules: {add^#(s(x), y) -> c_9(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
add^#(x1, x2) = [6] x1 + [7] x2 + [0]
c_8() = [1]
c_9(x1) = [1] x1 + [7]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_10() = [1]
* Path {12}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11() = [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [7]
p^#(x1) = [1] x1 + [7]
c_11() = [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ cond1(true(), x, y, z) -> cond2(gr(x, 0()), x, y, z)
, cond2(true(), x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
, cond2(false(), x, y, z) -> cond3(gr(y, 0()), x, y, z)
, cond3(true(), x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
, cond3(false(), x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond1^#(true(), x, y, z) -> c_0(cond2^#(gr(x, 0()), x, y, z))
, 2: cond2^#(true(), x, y, z) ->
c_1(cond1^#(gr(add(x, y), z), p(x), y, z))
, 3: cond2^#(false(), x, y, z) -> c_2(cond3^#(gr(y, 0()), x, y, z))
, 4: cond3^#(true(), x, y, z) ->
c_3(cond1^#(gr(add(x, y), z), x, p(y), z))
, 5: cond3^#(false(), x, y, z) ->
c_4(cond1^#(gr(add(x, y), z), x, y, z))
, 6: gr^#(0(), x) -> c_5()
, 7: gr^#(s(x), 0()) -> c_6()
, 8: gr^#(s(x), s(y)) -> c_7(gr^#(x, y))
, 9: add^#(0(), x) -> c_8(x)
, 10: add^#(s(x), y) -> c_9(add^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^2)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^2)) ]
->{8} [ YES(?,O(n^2)) ]
|
|->{6} [ YES(?,O(n^2)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{1,5,3,4,2} [ NA ]
Sub-problems:
-------------
* Path {1,5,3,4,2}: NA
--------------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 1] x1 + [0 1] x2 + [0]
[3 0] [3 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [3 0] x1 + [0]
[0 0] [0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
gr^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_7(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {8}->{6}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [3 0] x1 + [0]
[0 0] [0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_5()}
Weak Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
gr^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_5() = [1]
[0]
c_7(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {8}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [3 0] x1 + [0]
[0 0] [0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_6()}
Weak Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
gr^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_6() = [1]
[0]
c_7(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {10}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 1] x1 + [3 3] x2 + [0]
[3 0] [3 3] [0]
c_8(x1) = [3 0] x1 + [0]
[0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_9(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_9(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {10}->{9}: YES(?,O(n^2))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [1 1] x1 + [0]
[0 0] [0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_8(x)}
Weak Rules: {add^#(s(x), y) -> c_9(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [1]
add^#(x1, x2) = [3 2] x1 + [4 0] x2 + [0]
[2 5] [4 0] [0]
c_8(x1) = [3 0] x1 + [1]
[0 0] [0]
c_9(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [3 0] x1 + [0]
[0 0] [0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_10() = [0]
[1]
* Path {12}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
true() = [0]
[0]
cond2(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cond3(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
s(x1) = [1 1] x1 + [0]
[0 1] [0]
cond1^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
cond2^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_1(x1) = [3 0] x1 + [0]
[0 0] [0]
c_2(x1) = [3 0] x1 + [0]
[0 0] [0]
cond3^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [3 0] x1 + [0]
[0 0] [0]
c_9(x1) = [3 0] x1 + [0]
[0 0] [0]
p^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 0] [0]
p^#(x1) = [2 0] x1 + [7]
[2 0] [7]
c_11(x1) = [1 0] x1 + [0]
[0 0] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond1^#(true(), x, y, z) -> c_0(cond2^#(gr(x, 0()), x, y, z))
, 2: cond2^#(true(), x, y, z) ->
c_1(cond1^#(gr(add(x, y), z), p(x), y, z))
, 3: cond2^#(false(), x, y, z) -> c_2(cond3^#(gr(y, 0()), x, y, z))
, 4: cond3^#(true(), x, y, z) ->
c_3(cond1^#(gr(add(x, y), z), x, p(y), z))
, 5: cond3^#(false(), x, y, z) ->
c_4(cond1^#(gr(add(x, y), z), x, y, z))
, 6: gr^#(0(), x) -> c_5()
, 7: gr^#(s(x), 0()) -> c_6()
, 8: gr^#(s(x), s(y)) -> c_7(gr^#(x, y))
, 9: add^#(0(), x) -> c_8(x)
, 10: add^#(s(x), y) -> c_9(add^#(x, y))
, 11: p^#(0()) -> c_10()
, 12: p^#(s(x)) -> c_11(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{12} [ YES(?,O(n^1)) ]
->{11} [ YES(?,O(1)) ]
->{10} [ YES(?,O(n^1)) ]
|
`->{9} [ YES(?,O(n^1)) ]
->{8} [ YES(?,O(n^1)) ]
|
|->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{1,5,3,4,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,5,3,4,2}: MAYBE
-----------------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, p(0()) -> 0()
, p(s(x)) -> x}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ cond1^#(true(), x, y, z) -> c_0(cond2^#(gr(x, 0()), x, y, z))
, cond3^#(false(), x, y, z) ->
c_4(cond1^#(gr(add(x, y), z), x, y, z))
, cond2^#(false(), x, y, z) -> c_2(cond3^#(gr(y, 0()), x, y, z))
, cond3^#(true(), x, y, z) ->
c_3(cond1^#(gr(add(x, y), z), x, p(y), z))
, cond2^#(true(), x, y, z) ->
c_1(cond1^#(gr(add(x, y), z), p(x), y, z))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))
, p(0()) -> 0()
, p(s(x)) -> x}
Proof Output:
The input cannot be shown compatible
* Path {8}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [1] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [3] x1 + [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7(x1) = [1] x1 + [7]
* Path {8}->{6}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [3] x1 + [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_5()}
Weak Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_5) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_5() = [1]
c_7(x1) = [1] x1 + [2]
* Path {8}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [3] x1 + [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_6()}
Weak Rules: {gr^#(s(x), s(y)) -> c_7(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6() = [1]
c_7(x1) = [1] x1 + [7]
* Path {10}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [3] x1 + [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_9(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
add^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_9(x1) = [1] x1 + [7]
* Path {10}->{9}: YES(?,O(n^1))
-----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_8(x)}
Weak Rules: {add^#(s(x), y) -> c_9(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_8) = {1},
Uargs(c_9) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
add^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_8(x1) = [1] x1 + [1]
c_9(x1) = [1] x1 + [0]
* Path {11}: YES(?,O(1))
----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [0] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [3] x1 + [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_10()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_10() = [1]
* Path {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond1) = {}, Uargs(true) = {}, Uargs(cond2) = {},
Uargs(gr) = {}, Uargs(0) = {}, Uargs(add) = {}, Uargs(p) = {},
Uargs(false) = {}, Uargs(cond3) = {}, Uargs(s) = {},
Uargs(cond1^#) = {}, Uargs(c_0) = {}, Uargs(cond2^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(cond3^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(gr^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(add^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(p^#) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond1(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
true() = [0]
cond2(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
p(x1) = [0] x1 + [0]
false() = [0]
cond3(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
s(x1) = [1] x1 + [0]
cond1^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_0(x1) = [3] x1 + [0]
cond2^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_1(x1) = [3] x1 + [0]
c_2(x1) = [3] x1 + [0]
cond3^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_3(x1) = [3] x1 + [0]
c_4(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6() = [0]
c_7(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [3] x1 + [0]
c_9(x1) = [3] x1 + [0]
p^#(x1) = [3] x1 + [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [5]
p^#(x1) = [3] x1 + [0]
c_11(x1) = [3] x1 + [0]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.