Tool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> 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: minus^#(0(), Y) -> c_0()
, 2: minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))
, 3: geq^#(X, 0()) -> c_2()
, 4: geq^#(0(), s(Y)) -> c_3()
, 5: geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))
, 6: div^#(0(), s(Y)) -> c_5()
, 7: div^#(s(X), s(Y)) ->
c_6(if^#(geq(X, Y), s(div(minus(X, Y), s(Y))), 0()))
, 8: if^#(true(), X, Y) -> c_7()
, 9: if^#(false(), X, Y) -> c_8()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ NA ]
|
|->{8} [ NA ]
|
`->{9} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^2)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7() = [0]
[0]
c_8() = [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: {minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_1(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7() = [0]
[0]
c_8() = [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: {minus^#(0(), Y) -> c_0()}
Weak Rules: {minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
minus^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [7]
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7() = [0]
[0]
c_8() = [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: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
geq^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7() = [0]
[0]
c_8() = [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: {geq^#(X, 0()) -> c_2()}
Weak Rules: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 3] x1 + [2]
[0 1] [2]
geq^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
[0 2] [3 2] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7() = [0]
[0]
c_8() = [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: {geq^#(0(), s(Y)) -> c_3()}
Weak Rules: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
geq^#(x1, x2) = [2 3] x1 + [0 2] x2 + [0]
[0 3] [0 1] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7() = [0]
[0]
c_8() = [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: {div^#(0(), s(Y)) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [0 0] x1 + [2]
[0 0] [0]
div^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [3]
c_5() = [0]
[1]
* Path {7}: NA
------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {1}, Uargs(geq) = {},
Uargs(div) = {1}, Uargs(if) = {1, 2}, Uargs(minus^#) = {},
Uargs(c_1) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {},
Uargs(div^#) = {}, Uargs(c_6) = {1}, Uargs(if^#) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 2] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 3] x1 + [0]
[0 1] [3]
geq(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[1]
false() = [0]
[1]
div(x1, x2) = [1 3] x1 + [0 0] x2 + [1]
[0 3] [0 0] [0]
if(x1, x2, x3) = [2 1] x1 + [2 0] x2 + [2 0] x3 + [0]
[0 0] [0 1] [0 1] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [2 2] x1 + [2 1] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [2 0] x1 + [1 0] x2 + [0 0] x3 + [0]
[3 3] [3 3] [0 0] [0]
c_7() = [0]
[0]
c_8() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {7}->{8}: NA
-----------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {1}, Uargs(geq) = {},
Uargs(div) = {1}, Uargs(if) = {1, 2}, Uargs(minus^#) = {},
Uargs(c_1) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {},
Uargs(div^#) = {}, Uargs(c_6) = {1}, Uargs(if^#) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 3] x1 + [0]
[0 1] [3]
geq(x1, x2) = [0 2] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[1]
false() = [0]
[1]
div(x1, x2) = [1 3] x1 + [0 0] x2 + [3]
[0 1] [0 0] [0]
if(x1, x2, x3) = [2 1] x1 + [2 0] x2 + [1 0] x3 + [0]
[0 0] [0 1] [0 2] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7() = [0]
[0]
c_8() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {7}->{9}: NA
-----------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {1}, Uargs(geq) = {},
Uargs(div) = {1}, Uargs(if) = {1, 2}, Uargs(minus^#) = {},
Uargs(c_1) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {},
Uargs(div^#) = {}, Uargs(c_6) = {1}, Uargs(if^#) = {1, 2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 3] x1 + [0]
[0 1] [3]
geq(x1, x2) = [0 2] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[1]
false() = [0]
[1]
div(x1, x2) = [1 3] x1 + [0 0] x2 + [3]
[0 1] [0 0] [0]
if(x1, x2, x3) = [2 1] x1 + [2 0] x2 + [1 0] x3 + [0]
[0 0] [0 1] [0 2] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7() = [0]
[0]
c_8() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(0(), Y) -> c_0()
, 2: minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))
, 3: geq^#(X, 0()) -> c_2()
, 4: geq^#(0(), s(Y)) -> c_3()
, 5: geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))
, 6: div^#(0(), s(Y)) -> c_5()
, 7: div^#(s(X), s(Y)) ->
c_6(if^#(geq(X, Y), s(div(minus(X, Y), s(Y))), 0()))
, 8: if^#(true(), X, Y) -> c_7()
, 9: if^#(false(), X, Y) -> c_8()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
|->{8} [ NA ]
|
`->{9} [ MAYBE ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7() = [0]
c_8() = [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: {minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7() = [0]
c_8() = [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: {minus^#(0(), Y) -> c_0()}
Weak Rules: {minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_0() = [1]
c_1(x1) = [1] x1 + [5]
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
geq^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7() = [0]
c_8() = [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: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
geq^#(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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7() = [0]
c_8() = [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: {geq^#(X, 0()) -> c_2()}
Weak Rules: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
geq^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_2() = [1]
c_4(x1) = [1] x1 + [6]
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7() = [0]
c_8() = [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: {geq^#(0(), s(Y)) -> c_3()}
Weak Rules: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
geq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3() = [1]
c_4(x1) = [1] x1 + [7]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7() = [0]
c_8() = [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: {div^#(0(), s(Y)) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [0] x1 + [2]
div^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_5() = [0]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}.
* Path {7}->{8}: NA
-----------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {7}->{9}: MAYBE
--------------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> 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:
{ div^#(s(X), s(Y)) ->
c_6(if^#(geq(X, Y), s(div(minus(X, Y), s(Y))), 0()))
, if^#(false(), X, Y) -> c_8()
, minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
Proof Output:
The input cannot be shown compatible
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:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> 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: minus^#(0(), Y) -> c_0()
, 2: minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))
, 3: geq^#(X, 0()) -> c_2()
, 4: geq^#(0(), s(Y)) -> c_3()
, 5: geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))
, 6: div^#(0(), s(Y)) -> c_5()
, 7: div^#(s(X), s(Y)) ->
c_6(if^#(geq(X, Y), s(div(minus(X, Y), s(Y))), 0()))
, 8: if^#(true(), X, Y) -> c_7(X)
, 9: if^#(false(), X, Y) -> c_8(Y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
|->{8} [ NA ]
|
`->{9} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^2)) ]
|
|->{3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^2)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_1(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {minus^#(0(), Y) -> c_0()}
Weak Rules: {minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
minus^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [7]
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
geq^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_4(x1) = [1 0] x1 + [1]
[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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {geq^#(X, 0()) -> c_2()}
Weak Rules: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 3] x1 + [2]
[0 1] [2]
geq^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
[0 2] [3 2] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {geq^#(0(), s(Y)) -> c_3()}
Weak Rules: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
geq^#(x1, x2) = [2 3] x1 + [0 2] x2 + [0]
[0 3] [0 1] [0]
c_3() = [1]
[0]
c_4(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
div(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]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
geq^#(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]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(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_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(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: {div^#(0(), s(Y)) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [0 0] x1 + [2]
[0 0] [0]
div^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [3]
c_5() = [0]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}.
* Path {7}->{8}: NA
-----------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {7}->{9}: NA
-----------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(0(), Y) -> c_0()
, 2: minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))
, 3: geq^#(X, 0()) -> c_2()
, 4: geq^#(0(), s(Y)) -> c_3()
, 5: geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))
, 6: div^#(0(), s(Y)) -> c_5()
, 7: div^#(s(X), s(Y)) ->
c_6(if^#(geq(X, Y), s(div(minus(X, Y), s(Y))), 0()))
, 8: if^#(true(), X, Y) -> c_7(X)
, 9: if^#(false(), X, Y) -> c_8(Y)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
|->{8} [ MAYBE ]
|
`->{9} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {minus^#(0(), Y) -> c_0()}
Weak Rules: {minus^#(s(X), s(Y)) -> c_1(minus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_0() = [1]
c_1(x1) = [1] x1 + [5]
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
geq^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
geq^#(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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {geq^#(X, 0()) -> c_2()}
Weak Rules: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
geq^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_2() = [1]
c_4(x1) = [1] x1 + [6]
* 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(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {1}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {geq^#(0(), s(Y)) -> c_3()}
Weak Rules: {geq^#(s(X), s(Y)) -> c_4(geq^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(geq^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
geq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3() = [1]
c_4(x1) = [1] x1 + [7]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(geq) = {}, Uargs(div) = {},
Uargs(if) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(geq^#) = {}, Uargs(c_4) = {}, Uargs(div^#) = {},
Uargs(c_6) = {}, Uargs(if^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
geq(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
geq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(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: {div^#(0(), s(Y)) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(div^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [0] x1 + [2]
div^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_5() = [0]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{9}.
* Path {7}->{8}: MAYBE
--------------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> 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:
{ div^#(s(X), s(Y)) ->
c_6(if^#(geq(X, Y), s(div(minus(X, Y), s(Y))), 0()))
, if^#(true(), X, Y) -> c_7(X)
, minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
Proof Output:
The input cannot be shown compatible
* Path {7}->{9}: NA
-----------------
The usable rules for this path are:
{ minus(0(), Y) -> 0()
, minus(s(X), s(Y)) -> minus(X, Y)
, geq(X, 0()) -> true()
, geq(0(), s(Y)) -> false()
, geq(s(X), s(Y)) -> geq(X, Y)
, div(0(), s(Y)) -> 0()
, div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, if(true(), X, Y) -> X
, if(false(), X, Y) -> Y}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
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.