Tool CaT
stdout:
MAYBE
Problem:
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
YES(?,O(n^2))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, log(s(0())) -> 0()
, log(s(s(X))) -> s(log(s(quot(X, s(s(0()))))))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, log(s(0())) -> 0()
, log(s(s(X))) -> s(log(s(quot(X, s(s(0()))))))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: min^#(X, 0()) -> c_0()
, 2: min^#(s(X), s(Y)) -> c_1(min^#(X, Y))
, 3: quot^#(0(), s(Y)) -> c_2()
, 4: quot^#(s(X), s(Y)) -> c_3(quot^#(min(X, Y), s(Y)))
, 5: log^#(s(0())) -> c_4()
, 6: log^#(s(s(X))) -> c_5(log^#(s(quot(X, s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ 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(min) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(log) = {},
Uargs(min^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {}, Uargs(log^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(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]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(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]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {min^#(s(X), s(Y)) -> c_1(min^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
min^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_1(x1) = [1 2] x1 + [5]
[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(min) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(log) = {},
Uargs(min^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
Uargs(c_3) = {}, Uargs(log^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(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]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(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]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {min^#(X, 0()) -> c_0()}
Weak Rules: {min^#(s(X), s(Y)) -> c_1(min^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
min^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(log) = {},
Uargs(min^#) = {}, Uargs(c_1) = {}, Uargs(quot^#) = {1},
Uargs(c_3) = {1}, Uargs(log^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 1] x1 + [0 0] x2 + [3]
[0 1] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 1] x1 + [0]
[0 1] [1]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quot^#(s(X), s(Y)) -> c_3(quot^#(min(X, Y), s(Y)))}
Weak Rules:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
Proof Output:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 4] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
quot^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [0]
c_3(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(log) = {},
Uargs(min^#) = {}, Uargs(c_1) = {}, Uargs(quot^#) = {1},
Uargs(c_3) = {1}, Uargs(log^#) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
[0 2] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(Y)) -> c_2()}
Weak Rules:
{ quot^#(s(X), s(Y)) -> c_3(quot^#(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
Proof Output:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[4 4] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [1]
[0 0] [2]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [2 1] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {6}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(quot) = {1},
Uargs(log) = {}, Uargs(min^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(log^#) = {1},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 0] [0]
0() = [0]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [2]
quot(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(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]
log^#(x1) = [1 0] x1 + [0]
[3 3] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(s(s(X))) -> c_5(log^#(s(quot(X, s(s(0()))))))}
Weak Rules:
{ quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
Proof Output:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(log^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [2 2] x2 + [0]
[0 1] [0 0] [0]
0() = [2]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [4]
quot(x1, x2) = [0 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
log^#(x1) = [0 1] x1 + [0]
[4 0] [0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {6}->{5}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(quot) = {1},
Uargs(log) = {}, Uargs(min^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(log^#) = {1},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 1] [0 0] [0]
0() = [1]
[0]
s(x1) = [1 2] x1 + [0]
[0 1] [3]
quot(x1, x2) = [2 3] x1 + [3 2] x2 + [1]
[0 1] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(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]
log^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {log^#(s(0())) -> c_4()}
Weak Rules:
{ log^#(s(s(X))) -> c_5(log^#(s(quot(X, s(s(0()))))))
, quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
Proof Output:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(log^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[4 4] [4 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
quot(x1, x2) = [4 0] x1 + [0 0] x2 + [4]
[0 0] [0 0] [0]
log^#(x1) = [0 0] x1 + [2]
[0 0] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
YES(?,O(n^2))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^2))
Input Problem: runtime-complexity with respect to
Rules:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, log(s(0())) -> 0()
, log(s(s(X))) -> s(log(s(quot(X, s(s(0()))))))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^2))
Input Problem: runtime-complexity with respect to
Rules:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)
, quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, log(s(0())) -> 0()
, log(s(s(X))) -> s(log(s(quot(X, s(s(0()))))))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: min^#(X, 0()) -> c_0(X)
, 2: min^#(s(X), s(Y)) -> c_1(min^#(X, Y))
, 3: quot^#(0(), s(Y)) -> c_2()
, 4: quot^#(s(X), s(Y)) -> c_3(quot^#(min(X, Y), s(Y)))
, 5: log^#(s(0())) -> c_4()
, 6: log^#(s(s(X))) -> c_5(log^#(s(quot(X, s(s(0()))))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ 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(min) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(log) = {},
Uargs(min^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(log^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(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]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(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]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(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: {min^#(s(X), s(Y)) -> c_1(min^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
min^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_1(x1) = [1 2] x1 + [5]
[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(min) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(log) = {},
Uargs(min^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(log^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(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]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(x1, x2) = [3 3] x1 + [0 0] 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]
quot^#(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]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(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: {min^#(X, 0()) -> c_0(X)}
Weak Rules: {min^#(s(X), s(Y)) -> c_1(min^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
min^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
[4 1] [3 2] [0]
c_0(x1) = [0 0] x1 + [1]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[2 0] [0]
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(log) = {},
Uargs(min^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(log^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 1] x1 + [0 0] x2 + [3]
[0 1] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 1] x1 + [0]
[0 1] [1]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quot^#(s(X), s(Y)) -> c_3(quot^#(min(X, Y), s(Y)))}
Weak Rules:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
Proof Output:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 4] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
quot^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [0]
c_3(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(log) = {},
Uargs(min^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(log^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
[0 2] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
log^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(Y)) -> c_2()}
Weak Rules:
{ quot^#(s(X), s(Y)) -> c_3(quot^#(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
Proof Output:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[4 4] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [1]
[0 0] [2]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [2 1] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {6}: YES(?,O(n^2))
-----------------------
The usable rules for this path are:
{ quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {1}, Uargs(s) = {1}, Uargs(quot) = {1},
Uargs(log) = {}, Uargs(min^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(log^#) = {1}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [1]
[0 1] [2]
quot(x1, x2) = [1 2] x1 + [0 0] x2 + [3]
[0 1] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(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]
log^#(x1) = [2 0] x1 + [0]
[3 3] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(s(s(X))) -> c_5(log^#(s(quot(X, s(s(0()))))))}
Weak Rules:
{ quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
Proof Output:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(log^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [2 2] x2 + [0]
[0 1] [0 0] [0]
0() = [2]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [4]
quot(x1, x2) = [0 3] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
log^#(x1) = [0 1] x1 + [0]
[4 0] [0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules for this path are:
{ quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(min) = {1}, Uargs(s) = {1}, Uargs(quot) = {1},
Uargs(log) = {}, Uargs(min^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(log^#) = {1}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [2]
quot(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 2] [0 0] [0]
log(x1) = [0 0] x1 + [0]
[0 0] [0]
min^#(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]
quot^#(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]
log^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {log^#(s(0())) -> c_4()}
Weak Rules:
{ log^#(s(s(X))) -> c_5(log^#(s(quot(X, s(s(0()))))))
, quot(0(), s(Y)) -> 0()
, quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
, min(X, 0()) -> X
, min(s(X), s(Y)) -> min(X, Y)}
Proof Output:
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(log^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
min(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[4 4] [4 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
quot(x1, x2) = [4 0] x1 + [0 0] x2 + [4]
[0 0] [0 0] [0]
log^#(x1) = [0 0] x1 + [2]
[0 0] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]