Tool CaT
stdout:
MAYBE
Problem:
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(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:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(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: plus^#(0(), Y) -> c_0()
, 2: plus^#(s(X), Y) -> c_1(plus^#(X, Y))
, 3: min^#(X, 0()) -> c_2()
, 4: min^#(s(X), s(Y)) -> c_3(min^#(X, Y))
, 5: min^#(min(X, Y), Z()) -> c_4(min^#(X, plus(Y, Z())))
, 6: quot^#(0(), s(Y)) -> c_5()
, 7: quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ MAYBE ]
|
`->{6} [ NA ]
->{4,5} [ NA ]
|
`->{3} [ NA ]
->{2} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* 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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {1}, Uargs(min^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
min(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]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
min^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(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: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 1] [2]
plus^#(x1, x2) = [0 1 2] x1 + [0 0 0] x2 + [2]
[2 2 2] [4 4 4] [0]
[2 0 2] [4 4 4] [0]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* Path {2}->{1}: 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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {1}, Uargs(min^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min(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]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
min^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(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: {plus^#(0(), Y) -> c_0()}
Weak Rules: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 0] x1 + [0]
[0 1 3] [2]
[0 0 1] [2]
plus^#(x1, x2) = [0 3 2] x1 + [0 0 0] x2 + [0]
[2 2 2] [4 4 0] [0]
[2 2 2] [4 4 0] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {4,5}: NA
--------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0 0] x1 + [3 3 3] x2 + [2]
[2 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
min(x1, x2) = [1 2 1] x1 + [2 0 0] x2 + [0]
[0 0 0] [3 0 0] [0]
[0 0 0] [3 0 0] [0]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(x1, x2) = [2 3 1] x1 + [2 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(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4,5}->{3}: NA
-------------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0 0] x1 + [3 3 3] x2 + [2]
[0 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
min(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]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(x1, x2) = [0 0 0] x1 + [3 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(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: MAYBE
---------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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:
{ quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {2},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(quot^#) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 1] x1 + [2 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 2] [0 0 2] [0]
0() = [2]
[0]
[1]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [2]
min(x1, x2) = [2 3 2] x1 + [2 0 0] x2 + [0]
[0 3 0] [0 0 0] [3]
[0 0 2] [0 0 1] [0]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(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() = [0]
[0]
[0]
c_6(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: plus^#(0(), Y) -> c_0()
, 2: plus^#(s(X), Y) -> c_1(plus^#(X, Y))
, 3: min^#(X, 0()) -> c_2()
, 4: min^#(s(X), s(Y)) -> c_3(min^#(X, Y))
, 5: min^#(min(X, Y), Z()) -> c_4(min^#(X, plus(Y, Z())))
, 6: quot^#(0(), s(Y)) -> c_5()
, 7: quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ MAYBE ]
|
`->{6} [ NA ]
->{4,5} [ NA ]
|
`->{3} [ NA ]
->{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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {1}, Uargs(min^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(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]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(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]
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: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
plus^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[4 5] [4 4] [0]
c_1(x1) = [1 0] x1 + [1]
[0 0] [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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {1}, Uargs(min^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(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]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(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]
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: {plus^#(0(), Y) -> c_0()}
Weak Rules: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 3] x1 + [2]
[0 0] [2]
plus^#(x1, x2) = [3 0] x1 + [0 0] x2 + [4]
[2 2] [0 4] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {4,5}: NA
--------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [3 0] x1 + [3 3] x2 + [2]
[0 0] [3 3] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
min(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [2 0] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(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]
min^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(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]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4,5}->{3}: NA
-------------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0] x1 + [3 3] x2 + [2]
[0 0] [3 3] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(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]
min^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(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]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: MAYBE
---------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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:
{ quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {2},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(quot^#) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [1 3] x1 + [1 0] x2 + [0]
[0 1] [0 2] [0]
0() = [1]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [3]
min(x1, x2) = [1 3] x1 + [1 0] x2 + [2]
[0 2] [0 2] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(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]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [3 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]
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: plus^#(0(), Y) -> c_0()
, 2: plus^#(s(X), Y) -> c_1(plus^#(X, Y))
, 3: min^#(X, 0()) -> c_2()
, 4: min^#(s(X), s(Y)) -> c_3(min^#(X, Y))
, 5: min^#(min(X, Y), Z()) -> c_4(min^#(X, plus(Y, Z())))
, 6: quot^#(0(), s(Y)) -> c_5()
, 7: quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ MAYBE ]
|
`->{6} [ NA ]
->{4,5} [ NA ]
|
`->{3} [ NA ]
->{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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {1}, Uargs(min^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(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: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
plus^#(x1, x2) = [2] x1 + [7] 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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_1) = {1}, Uargs(min^#) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(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: {plus^#(0(), Y) -> c_0()}
Weak Rules: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
plus^#(x1, x2) = [2] x1 + [7] x2 + [4]
c_0() = [1]
c_1(x1) = [1] x1 + [4]
* Path {4,5}: NA
--------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [3] x1 + [3] x2 + [3]
0() = [1]
s(x1) = [1] x1 + [2]
min(x1, x2) = [3] x1 + [3] x2 + [0]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4,5}->{3}: NA
-------------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2] x1 + [3] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [1]
min(x1, x2) = [0] x1 + [0] x2 + [0]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: MAYBE
---------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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:
{ quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {2},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {},
Uargs(min^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
Uargs(quot^#) = {1}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2] x1 + [2] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [1]
min(x1, x2) = [3] x1 + [2] x2 + [1]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_5() = [0]
c_6(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:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(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: plus^#(0(), Y) -> c_0(Y)
, 2: plus^#(s(X), Y) -> c_1(plus^#(X, Y))
, 3: min^#(X, 0()) -> c_2(X)
, 4: min^#(s(X), s(Y)) -> c_3(min^#(X, Y))
, 5: min^#(min(X, Y), Z()) -> c_4(min^#(X, plus(Y, Z())))
, 6: quot^#(0(), s(Y)) -> c_5()
, 7: quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ MAYBE ]
|
`->{6} [ NA ]
->{4,5} [ NA ]
|
`->{3} [ NA ]
->{2} [ YES(?,O(n^3)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* 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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [1 3 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
min(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]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
min^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(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: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 4] x1 + [2]
[0 1 2] [2]
[0 0 1] [2]
plus^#(x1, x2) = [0 1 2] x1 + [0 0 0] x2 + [2]
[2 2 2] [4 4 4] [0]
[2 0 2] [4 4 4] [0]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* 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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min(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]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
min^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(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: {plus^#(0(), Y) -> c_0(Y)}
Weak Rules: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 2 3] x1 + [0]
[0 0 3] [2]
[0 0 1] [2]
plus^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [2]
[0 2 2] [0 0 0] [0]
[4 0 0] [4 0 4] [4]
c_0(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [5]
[0 0 0] [7]
[2 2 0] [0]
* Path {4,5}: NA
--------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {2}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0 0] x1 + [3 3 3] x2 + [2]
[2 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 1 0] [0]
[0 0 1] [0]
min(x1, x2) = [1 2 1] x1 + [2 0 0] x2 + [0]
[0 0 0] [3 0 0] [0]
[0 0 0] [3 0 0] [0]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(x1, x2) = [2 3 1] x1 + [2 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4,5}->{3}: NA
-------------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {2}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [3 0 0] x1 + [3 3 3] x2 + [2]
[0 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [2]
[0 0 0] [0]
[0 0 0] [0]
min(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]
Z() = [0]
[0]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(x1, x2) = [3 3 3] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: MAYBE
---------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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:
{ quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {1}, Uargs(s) = {1}, Uargs(min) = {1, 2},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {1},
Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [1 2 0] x1 + [1 0 0] x2 + [2]
[0 1 1] [2 2 0] [0]
[0 0 0] [0 0 1] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [1]
[0 0 1] [0]
min(x1, x2) = [1 2 0] x1 + [1 0 1] x2 + [3]
[3 1 0] [0 1 0] [3]
[0 0 1] [0 0 0] [0]
Z() = [3]
[1]
[0]
quot(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]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
min^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(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() = [0]
[0]
[0]
c_6(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: plus^#(0(), Y) -> c_0(Y)
, 2: plus^#(s(X), Y) -> c_1(plus^#(X, Y))
, 3: min^#(X, 0()) -> c_2(X)
, 4: min^#(s(X), s(Y)) -> c_3(min^#(X, Y))
, 5: min^#(min(X, Y), Z()) -> c_4(min^#(X, plus(Y, Z())))
, 6: quot^#(0(), s(Y)) -> c_5()
, 7: quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ MAYBE ]
|
`->{6} [ NA ]
->{4,5} [ NA ]
|
`->{3} [ NA ]
->{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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(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]
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: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [2]
plus^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[4 5] [4 4] [0]
c_1(x1) = [1 0] x1 + [1]
[0 0] [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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(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]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(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]
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: {plus^#(0(), Y) -> c_0(Y)}
Weak Rules: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
plus^#(x1, x2) = [2 0] x1 + [0 0] x2 + [4]
[2 2] [4 4] [0]
c_0(x1) = [0 0] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [4]
* Path {4,5}: NA
--------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {2}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [3 0] x1 + [3 3] x2 + [2]
[0 0] [3 3] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
min(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [2 0] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(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]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4,5}->{3}: NA
-------------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {2}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0] x1 + [3 3] x2 + [2]
[0 0] [3 3] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
Z() = [0]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [3 3] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [1 1] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(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]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: MAYBE
---------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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:
{ quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {1}, Uargs(s) = {1}, Uargs(min) = {1, 2},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {1},
Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 1] [0]
0() = [2]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
min(x1, x2) = [1 1] x1 + [1 0] x2 + [2]
[0 2] [1 0] [1]
Z() = [1]
[0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [3 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]
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: plus^#(0(), Y) -> c_0(Y)
, 2: plus^#(s(X), Y) -> c_1(plus^#(X, Y))
, 3: min^#(X, 0()) -> c_2(X)
, 4: min^#(s(X), s(Y)) -> c_3(min^#(X, Y))
, 5: min^#(min(X, Y), Z()) -> c_4(min^#(X, plus(Y, Z())))
, 6: quot^#(0(), s(Y)) -> c_5()
, 7: quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ MAYBE ]
|
`->{6} [ NA ]
->{4,5} [ NA ]
|
`->{3} [ NA ]
->{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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(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: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
plus^#(x1, x2) = [2] x1 + [7] 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(plus) = {}, Uargs(s) = {}, Uargs(min) = {}, Uargs(quot) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(min^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
min(x1, x2) = [0] x1 + [0] x2 + [0]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(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: {plus^#(0(), Y) -> c_0(Y)}
Weak Rules: {plus^#(s(X), Y) -> c_1(plus^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
plus^#(x1, x2) = [3] x1 + [7] x2 + [2]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [5]
* Path {4,5}: NA
--------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {2}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [3] x1 + [3] x2 + [3]
0() = [1]
s(x1) = [1] x1 + [2]
min(x1, x2) = [3] x1 + [3] x2 + [0]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4,5}->{3}: NA
-------------------
The usable rules for this path are:
{ plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {1}, Uargs(min) = {},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {2}, Uargs(c_2) = {},
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
Uargs(c_6) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2] x1 + [3] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [1]
min(x1, x2) = [0] x1 + [0] x2 + [0]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: MAYBE
---------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(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:
{ quot^#(s(X), s(Y)) -> c_6(quot^#(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
Proof Output:
The input cannot be shown compatible
* Path {7}->{6}: NA
-----------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, min(min(X, Y), Z()) -> min(X, plus(Y, Z()))
, plus(0(), Y) -> Y
, plus(s(X), Y) -> s(plus(X, Y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {1}, Uargs(s) = {1}, Uargs(min) = {1, 2},
Uargs(quot) = {}, Uargs(plus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {1},
Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2] x1 + [2] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [2]
min(x1, x2) = [2] x1 + [1] x2 + [1]
Z() = [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
min^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_5() = [0]
c_6(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.