Tool CaT
stdout:
MAYBE
Problem:
p(0()) -> 0()
p(s(x)) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,y) -> if(le(x,y),x,y)
if(true(),x,y) -> 0()
if(false(),x,y) -> s(minus(p(x),y))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, y) -> if(le(x, y), x, y)
, if(true(), x, y) -> 0()
, if(false(), x, y) -> s(minus(p(x), y))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: p^#(0()) -> c_0()
, 2: p^#(s(x)) -> c_1()
, 3: le^#(0(), y) -> c_2()
, 4: le^#(s(x), 0()) -> c_3()
, 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
, 6: minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, 7: if^#(true(), x, y) -> c_6()
, 8: if^#(false(), x, y) -> c_7(minus^#(p(x), y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,8} [ MAYBE ]
|
`->{7} [ NA ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^3)) ]
->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {}, Uargs(c_5) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_0()}
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_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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {}, Uargs(c_5) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_1()}
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_1() = [0]
[1]
[1]
* Path {5}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
le^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_4(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {5}->{3}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(0(), y) -> c_2()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
le^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {5}->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(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^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), 0()) -> c_3()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
le^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {6,8}: MAYBE
-----------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, if^#(false(), x, y) -> c_7(minus^#(p(x), y))
, p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6,8}->{7}: NA
-------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {1}, Uargs(c_5) = {1}, Uargs(if^#) = {1},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [2 0 0] x1 + [0]
[1 1 0] [2]
[0 0 1] [0]
0() = [2]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
le(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [1]
[0]
[0]
false() = [1]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(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 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: p^#(0()) -> c_0()
, 2: p^#(s(x)) -> c_1()
, 3: le^#(0(), y) -> c_2()
, 4: le^#(s(x), 0()) -> c_3()
, 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
, 6: minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, 7: if^#(true(), x, y) -> c_6()
, 8: if^#(false(), x, y) -> c_7(minus^#(p(x), y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,8} [ MAYBE ]
|
`->{7} [ NA ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {}, Uargs(c_5) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(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_0()}
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_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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {}, Uargs(c_5) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(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_1()}
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_1() = [0]
[1]
* Path {5}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_4(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {5}->{3}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(0(), y) -> c_2()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {5}->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), 0()) -> c_3()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_3() = [1]
[0]
c_4(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {6,8}: MAYBE
-----------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, if^#(false(), x, y) -> c_7(minus^#(p(x), y))
, p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6,8}->{7}: NA
-------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {1}, Uargs(c_5) = {1}, Uargs(if^#) = {1},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [2 1] x1 + [0]
[2 0] [1]
0() = [0]
[1]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
le(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [3]
true() = [0]
[1]
false() = [1]
[1]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(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: p^#(0()) -> c_0()
, 2: p^#(s(x)) -> c_1()
, 3: le^#(0(), y) -> c_2()
, 4: le^#(s(x), 0()) -> c_3()
, 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
, 6: minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, 7: if^#(true(), x, y) -> c_6()
, 8: if^#(false(), x, y) -> c_7(minus^#(p(x), y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,8} [ MAYBE ]
|
`->{7} [ NA ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {}, Uargs(c_5) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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_0()}
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_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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {}, Uargs(c_5) = {}, Uargs(if^#) = {},
Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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_1()}
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_1() = [1]
* Path {5}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{3}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(0(), y) -> c_2()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_2() = [1]
c_4(x1) = [1] x1 + [2]
* Path {5}->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), 0()) -> c_3()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3() = [1]
c_4(x1) = [1] x1 + [7]
* Path {6,8}: MAYBE
-----------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, if^#(false(), x, y) -> c_7(minus^#(p(x), y))
, p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6,8}->{7}: NA
-------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(le^#) = {}, Uargs(c_4) = {},
Uargs(minus^#) = {1}, Uargs(c_5) = {1}, Uargs(if^#) = {1},
Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [3] x1 + [3]
0() = [0]
s(x1) = [1] x1 + [2]
le(x1, x2) = [2] x1 + [0] x2 + [2]
true() = [1]
false() = [1]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, minus(x, y) -> if(le(x, y), x, y)
, if(true(), x, y) -> 0()
, if(false(), x, y) -> s(minus(p(x), y))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: p^#(0()) -> c_0()
, 2: p^#(s(x)) -> c_1(x)
, 3: le^#(0(), y) -> c_2()
, 4: le^#(s(x), 0()) -> c_3()
, 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
, 6: minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, 7: if^#(true(), x, y) -> c_6()
, 8: if^#(false(), x, y) -> c_7(minus^#(p(x), y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,8} [ MAYBE ]
|
`->{7} [ NA ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^3)) ]
->{2} [ YES(?,O(n^3)) ]
->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_0()}
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_0() = [0]
[1]
[1]
* Path {2}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 3 3] x1 + [0]
[0 1 1] [0]
[0 0 1] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [2]
[0 0 0] [2]
p^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {5}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(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: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
le^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_4(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {5}->{3}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(0(), y) -> c_2()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
le^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {5}->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(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_6() = [0]
[0]
[0]
c_7(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^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), 0()) -> c_3()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
le^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {6,8}: MAYBE
-----------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, if^#(false(), x, y) -> c_7(minus^#(p(x), y))
, p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6,8}->{7}: NA
-------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {1}, Uargs(s) = {}, Uargs(le) = {1}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {1}, Uargs(c_5) = {1},
Uargs(if^#) = {1, 2}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [2 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
0() = [2]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
le(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
true() = [1]
[0]
[0]
false() = [1]
[0]
[0]
minus(x1, 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]
p^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
le^#(x1, 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_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
minus^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
if^#(x1, x2, x3) = [3 0 0] x1 + [3 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_6() = [0]
[0]
[0]
c_7(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 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: p^#(0()) -> c_0()
, 2: p^#(s(x)) -> c_1(x)
, 3: le^#(0(), y) -> c_2()
, 4: le^#(s(x), 0()) -> c_3()
, 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
, 6: minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, 7: if^#(true(), x, y) -> c_6()
, 8: if^#(false(), x, y) -> c_7(minus^#(p(x), y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,8} [ MAYBE ]
|
`->{7} [ NA ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^2)) ]
->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(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_0()}
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_0() = [0]
[1]
* Path {2}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 1] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(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_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}
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_1(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {5}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_4(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {5}->{3}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(0(), y) -> c_2()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {5}->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(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_6() = [0]
[0]
c_7(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: {le^#(s(x), 0()) -> c_3()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_3() = [1]
[0]
c_4(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {6,8}: MAYBE
-----------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, if^#(false(), x, y) -> c_7(minus^#(p(x), y))
, p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6,8}->{7}: NA
-------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {1}, Uargs(s) = {}, Uargs(le) = {1}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {1}, Uargs(c_5) = {1},
Uargs(if^#) = {1, 2}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [2 0] x1 + [0]
[0 1] [0]
0() = [2]
[0]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
le(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [3]
true() = [1]
[1]
false() = [1]
[1]
minus(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
minus^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [3 0] x1 + [3 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: p^#(0()) -> c_0()
, 2: p^#(s(x)) -> c_1(x)
, 3: le^#(0(), y) -> c_2()
, 4: le^#(s(x), 0()) -> c_3()
, 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
, 6: minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, 7: if^#(true(), x, y) -> c_6()
, 8: if^#(false(), x, y) -> c_7(minus^#(p(x), y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6,8} [ MAYBE ]
|
`->{7} [ NA ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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_0()}
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_0() = [1]
* Path {2}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [3] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [7]
p^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [1]
* Path {5}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{3}: 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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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: {le^#(0(), y) -> c_2()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_2() = [1]
c_4(x1) = [1] x1 + [2]
* Path {5}->{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(p) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {1}, Uargs(minus^#) = {}, Uargs(c_5) = {},
Uargs(if^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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: {le^#(s(x), 0()) -> c_3()}
Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3() = [1]
c_4(x1) = [1] x1 + [7]
* Path {6,8}: MAYBE
-----------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ minus^#(x, y) -> c_5(if^#(le(x, y), x, y))
, if^#(false(), x, y) -> c_7(minus^#(p(x), y))
, p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {6,8}->{7}: NA
-------------------
The usable rules for this path are:
{ p(0()) -> 0()
, p(s(x)) -> x
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(p) = {1}, Uargs(s) = {}, Uargs(le) = {1}, Uargs(minus) = {},
Uargs(if) = {}, Uargs(p^#) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
Uargs(c_4) = {}, Uargs(minus^#) = {1}, Uargs(c_5) = {1},
Uargs(if^#) = {1, 2}, Uargs(c_7) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
p(x1) = [3] x1 + [3]
0() = [0]
s(x1) = [1] x1 + [2]
le(x1, x2) = [2] x1 + [0] x2 + [1]
true() = [0]
false() = [0]
minus(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
p^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_5(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [3] x1 + [3] x2 + [0] x3 + [0]
c_6() = [0]
c_7(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.