Tool CaT
stdout:
MAYBE
Problem:
f(tt(),x) -> f(eq(x,half(double(x))),s(x))
eq(s(x),s(y)) -> eq(x,y)
eq(0(),0()) -> tt()
double(s(x)) -> s(s(double(x)))
double(0()) -> 0()
half(s(s(x))) -> s(half(x))
half(0()) -> 0()
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:
{ f(tt(), x) -> f(eq(x, half(double(x))), s(x))
, eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
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: f^#(tt(), x) -> c_0(f^#(eq(x, half(double(x))), s(x)))
, 2: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
, 3: eq^#(0(), 0()) -> c_2()
, 4: double^#(s(x)) -> c_3(double^#(x))
, 5: double^#(0()) -> c_4()
, 6: half^#(s(s(x))) -> c_5(half^#(x))
, 7: half^#(0()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
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:
{ f^#(tt(), x) -> c_0(f^#(eq(x, half(double(x))), s(x)))
, eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [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: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
eq^#(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}->{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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [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: {eq^#(0(), 0()) -> c_2()}
Weak Rules: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [0]
0() = [2]
[2]
eq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[2 0] [0 0] [4]
c_1(x1) = [1 0] x1 + [7]
[0 0] [7]
c_2() = [1]
[0]
* Path {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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
double^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [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: {double^#(s(x)) -> c_3(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
double^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {4}->{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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [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: {double^#(0()) -> c_4()}
Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [3]
0() = [2]
[2]
double^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_3(x1) = [1 0] x1 + [5]
[2 0] [3]
c_4() = [1]
[0]
* Path {6}: 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [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: {half^#(s(s(x))) -> c_5(half^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
half^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_5(x1) = [1 0] x1 + [5]
[2 0] [7]
* Path {6}->{7}: 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [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: {half^#(0()) -> c_6()}
Weak Rules: {half^#(s(s(x))) -> c_5(half^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [0]
0() = [0]
[2]
half^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_5(x1) = [1 0] x1 + [3]
[2 0] [3]
c_6() = [1]
[0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(tt(), x) -> c_0(f^#(eq(x, half(double(x))), s(x)))
, 2: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
, 3: eq^#(0(), 0()) -> c_2()
, 4: double^#(s(x)) -> c_3(double^#(x))
, 5: double^#(0()) -> c_4()
, 6: half^#(s(s(x))) -> c_5(half^#(x))
, 7: half^#(0()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
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:
{ f^#(tt(), x) -> c_0(f^#(eq(x, half(double(x))), s(x)))
, eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
eq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {eq^#(0(), 0()) -> c_2()}
Weak Rules: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [0]
eq^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1(x1) = [1] x1 + [3]
c_2() = [1]
* Path {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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
double^#(x1) = [3] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {double^#(s(x)) -> c_3(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
double^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {double^#(0()) -> c_4()}
Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
double^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [1]
* Path {6}: 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
half^#(x1) = [3] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [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: {half^#(s(s(x))) -> c_5(half^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
half^#(x1) = [2] x1 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{7}: 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [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: {half^#(0()) -> c_6()}
Weak Rules: {half^#(s(s(x))) -> c_5(half^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
half^#(x1) = [2] x1 + [4]
c_5(x1) = [1] x1 + [0]
c_6() = [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f(tt(), x) -> f(eq(x, half(double(x))), s(x))
, eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
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: f^#(tt(), x) -> c_0(f^#(eq(x, half(double(x))), s(x)))
, 2: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
, 3: eq^#(0(), 0()) -> c_2()
, 4: double^#(s(x)) -> c_3(double^#(x))
, 5: double^#(0()) -> c_4()
, 6: half^#(s(s(x))) -> c_5(half^#(x))
, 7: half^#(0()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
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:
{ f^#(tt(), x) -> c_0(f^#(eq(x, half(double(x))), s(x)))
, eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [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: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
eq^#(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}->{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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [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: {eq^#(0(), 0()) -> c_2()}
Weak Rules: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [0]
0() = [2]
[2]
eq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[2 0] [0 0] [4]
c_1(x1) = [1 0] x1 + [7]
[0 0] [7]
c_2() = [1]
[0]
* Path {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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
double^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [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: {double^#(s(x)) -> c_3(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
double^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {4}->{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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [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: {double^#(0()) -> c_4()}
Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [3]
0() = [2]
[2]
double^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_3(x1) = [1 0] x1 + [5]
[2 0] [3]
c_4() = [1]
[0]
* Path {6}: 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [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: {half^#(s(s(x))) -> c_5(half^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
half^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_5(x1) = [1 0] x1 + [5]
[2 0] [7]
* Path {6}->{7}: 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1) = [1 0] x1 + [0]
[0 1] [0]
c_6() = [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: {half^#(0()) -> c_6()}
Weak Rules: {half^#(s(s(x))) -> c_5(half^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [0]
0() = [0]
[2]
half^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_5(x1) = [1 0] x1 + [3]
[2 0] [3]
c_6() = [1]
[0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(tt(), x) -> c_0(f^#(eq(x, half(double(x))), s(x)))
, 2: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
, 3: eq^#(0(), 0()) -> c_2()
, 4: double^#(s(x)) -> c_3(double^#(x))
, 5: double^#(0()) -> c_4()
, 6: half^#(s(s(x))) -> c_5(half^#(x))
, 7: half^#(0()) -> c_6()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
`->{7} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
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:
{ f^#(tt(), x) -> c_0(f^#(eq(x, half(double(x))), s(x)))
, eq(s(x), s(y)) -> eq(x, y)
, eq(0(), 0()) -> tt()
, double(s(x)) -> s(s(double(x)))
, double(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, half(0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
eq^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {eq^#(0(), 0()) -> c_2()}
Weak Rules: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [0]
eq^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1(x1) = [1] x1 + [3]
c_2() = [1]
* Path {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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
double^#(x1) = [3] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {double^#(s(x)) -> c_3(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
double^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {double^#(0()) -> c_4()}
Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
double^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [1]
* Path {6}: 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
half^#(x1) = [3] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [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: {half^#(s(s(x))) -> c_5(half^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
half^#(x1) = [2] x1 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{7}: 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(f) = {}, Uargs(eq) = {}, Uargs(half) = {},
Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
tt() = [0]
eq(x1, x2) = [0] x1 + [0] x2 + [0]
half(x1) = [0] x1 + [0]
double(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
double^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
half^#(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [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: {half^#(0()) -> c_6()}
Weak Rules: {half^#(s(s(x))) -> c_5(half^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
half^#(x1) = [2] x1 + [4]
c_5(x1) = [1] x1 + [0]
c_6() = [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.