Tool CaT
stdout:
MAYBE
Problem:
f(true(),x,y) -> f(and(gt(x,y),gt(y,s(s(0())))),plus(s(0()),x),double(y))
gt(0(),v) -> false()
gt(s(u),0()) -> true()
gt(s(u),s(v)) -> gt(u,v)
and(x,true()) -> x
and(x,false()) -> false()
plus(n,0()) -> n
plus(n,s(m)) -> s(plus(n,m))
plus(plus(n,m),u) -> plus(n,plus(m,u))
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
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(true(), x, y) ->
f(and(gt(x, y), gt(y, s(s(0())))), plus(s(0()), x), double(y))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
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^#(true(), x, y) ->
c_0(f^#(and(gt(x, y), gt(y, s(s(0())))),
plus(s(0()), x),
double(y)))
, 2: gt^#(0(), v) -> c_1()
, 3: gt^#(s(u), 0()) -> c_2()
, 4: gt^#(s(u), s(v)) -> c_3(gt^#(u, v))
, 5: and^#(x, true()) -> c_4()
, 6: and^#(x, false()) -> c_5()
, 7: plus^#(n, 0()) -> c_6()
, 8: plus^#(n, s(m)) -> c_7(plus^#(n, m))
, 9: plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, 10: double^#(0()) -> c_9()
, 11: double^#(s(x)) -> c_10(double^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{8,9} [ inherited ]
|
`->{7} [ MAYBE ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [1 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
and^#(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_4() = [0]
[0]
[0]
c_5() = [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
gt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_3(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
and^#(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_4() = [0]
[0]
[0]
c_5() = [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_1()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
0() = [0]
[2]
[0]
gt^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 0] [0 4 0] [0]
c_1() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {4}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
and^#(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_4() = [0]
[0]
[0]
c_5() = [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), 0()) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0 0] x1 + [0]
[0 1 2] [2]
[0 0 0] [2]
0() = [0]
[0]
[0]
gt^#(x1, x2) = [0 2 2] x1 + [0 0 0] x2 + [0]
[0 0 0] [4 0 0] [4]
[0 0 0] [0 0 0] [2]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(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_4() = [0]
[0]
[0]
c_5() = [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
[2]
and^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_4() = [0]
[1]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(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_4() = [0]
[0]
[0]
c_5() = [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
and^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_5() = [0]
[1]
[1]
* Path {8,9}: inherited
---------------------
This path is subsumed by the proof of path {8,9}->{7}.
* Path {8,9}->{7}: MAYBE
----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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:
{ plus^#(n, s(m)) -> c_7(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_6()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
* Path {11}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(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_4() = [0]
[0]
[0]
c_5() = [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {double^#(s(x)) -> c_10(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
double^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_10(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {11}->{10}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(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_4() = [0]
[0]
[0]
c_5() = [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_6() = [0]
[0]
[0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {double^#(0()) -> c_9()}
Weak Rules: {double^#(s(x)) -> c_10(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
0() = [2]
[2]
[2]
double^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_9() = [1]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(true(), x, y) ->
c_0(f^#(and(gt(x, y), gt(y, s(s(0())))),
plus(s(0()), x),
double(y)))
, 2: gt^#(0(), v) -> c_1()
, 3: gt^#(s(u), 0()) -> c_2()
, 4: gt^#(s(u), s(v)) -> c_3(gt^#(u, v))
, 5: and^#(x, true()) -> c_4()
, 6: and^#(x, false()) -> c_5()
, 7: plus^#(n, 0()) -> c_6()
, 8: plus^#(n, s(m)) -> c_7(plus^#(n, m))
, 9: plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, 10: double^#(0()) -> c_9()
, 11: double^#(s(x)) -> c_10(double^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{8,9} [ inherited ]
|
`->{7} [ MAYBE ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^2)) ]
|
|->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(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: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_3(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(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: {gt^#(0(), v) -> c_1()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 1] [2]
0() = [2]
[2]
gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_1() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(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: {gt^#(s(u), 0()) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [2]
0() = [0]
[2]
gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_4() = [0]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_5() = [0]
[1]
* Path {8,9}: inherited
---------------------
This path is subsumed by the proof of path {8,9}->{7}.
* Path {8,9}->{7}: MAYBE
----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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:
{ plus^#(n, s(m)) -> c_7(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_6()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
* Path {11}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [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_10(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {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_10(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11}->{10}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [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_9()}
Weak Rules: {double^#(s(x)) -> c_10(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {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_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [5]
[2 0] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(true(), x, y) ->
c_0(f^#(and(gt(x, y), gt(y, s(s(0())))),
plus(s(0()), x),
double(y)))
, 2: gt^#(0(), v) -> c_1()
, 3: gt^#(s(u), 0()) -> c_2()
, 4: gt^#(s(u), s(v)) -> c_3(gt^#(u, v))
, 5: and^#(x, true()) -> c_4()
, 6: and^#(x, false()) -> c_5()
, 7: plus^#(n, 0()) -> c_6()
, 8: plus^#(n, s(m)) -> c_7(plus^#(n, m))
, 9: plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, 10: double^#(0()) -> c_9()
, 11: double^#(s(x)) -> c_10(double^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{8,9} [ inherited ]
|
`->{7} [ MAYBE ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(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: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(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: {gt^#(0(), v) -> c_1()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_1() = [1]
c_3(x1) = [1] x1 + [5]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(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: {gt^#(s(u), 0()) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2() = [1]
c_3(x1) = [1] x1 + [7]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_4() = [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_5() = [1]
* Path {8,9}: inherited
---------------------
This path is subsumed by the proof of path {8,9}->{7}.
* Path {8,9}->{7}: MAYBE
----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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:
{ plus^#(n, s(m)) -> c_7(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_6()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
* Path {11}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [3] x1 + [0]
c_9() = [0]
c_10(x1) = [1] 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: {double^#(s(x)) -> c_10(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
double^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [7]
* Path {11}->{10}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(plus^#) = {}, Uargs(c_7) = {},
Uargs(c_8) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [1] 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: {double^#(0()) -> c_9()}
Weak Rules: {double^#(s(x)) -> c_10(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
double^#(x1) = [2] x1 + [0]
c_9() = [1]
c_10(x1) = [1] x1 + [0]
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:
{ f(true(), x, y) ->
f(and(gt(x, y), gt(y, s(s(0())))), plus(s(0()), x), double(y))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
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^#(true(), x, y) ->
c_0(f^#(and(gt(x, y), gt(y, s(s(0())))),
plus(s(0()), x),
double(y)))
, 2: gt^#(0(), v) -> c_1()
, 3: gt^#(s(u), 0()) -> c_2()
, 4: gt^#(s(u), s(v)) -> c_3(gt^#(u, v))
, 5: and^#(x, true()) -> c_4(x)
, 6: and^#(x, false()) -> c_5()
, 7: plus^#(n, 0()) -> c_6(n)
, 8: plus^#(n, s(m)) -> c_7(plus^#(n, m))
, 9: plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, 10: double^#(0()) -> c_9()
, 11: double^#(s(x)) -> c_10(double^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{8,9} [ inherited ]
|
`->{7} [ MAYBE ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^2)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [1 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
and^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 3] x1 + [0]
[0 0 1] [4]
[0 0 0] [2]
gt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
[0 0 2] [0 0 0] [2]
[1 1 0] [4 1 0] [0]
c_3(x1) = [1 2 0] x1 + [3]
[0 0 0] [2]
[0 2 0] [3]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
and^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(0(), v) -> c_1()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 0] [0]
0() = [0]
[2]
[0]
gt^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[2 0 0] [0 4 0] [0]
c_1() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {4}->{3}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
and^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(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: {gt^#(s(u), 0()) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0 0] x1 + [0]
[0 1 2] [2]
[0 0 0] [2]
0() = [0]
[0]
[0]
gt^#(x1, x2) = [0 2 2] x1 + [0 0 0] x2 + [0]
[0 0 0] [4 0 0] [4]
[0 0 0] [0 0 0] [2]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[0]
[2]
and^#(x1, x2) = [7 7 7] x1 + [2 0 2] x2 + [7]
[7 7 7] [2 0 2] [7]
[7 7 7] [2 0 2] [7]
c_4(x1) = [1 3 3] x1 + [0]
[1 1 1] [1]
[1 1 1] [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
[2]
and^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
[0 0 0] [2 2 0] [3]
[0 0 0] [2 2 2] [3]
c_5() = [0]
[1]
[1]
* Path {8,9}: inherited
---------------------
This path is subsumed by the proof of path {8,9}->{7}.
* Path {8,9}->{7}: MAYBE
----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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:
{ plus^#(n, s(m)) -> c_7(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_6(n)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
* Path {11}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [1 3 0] x1 + [0]
[0 1 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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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: {double^#(s(x)) -> c_10(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 1 2] [2]
[0 0 0] [0]
double^#(x1) = [0 1 0] x1 + [2]
[6 0 0] [0]
[2 3 0] [2]
c_10(x1) = [1 0 0] x1 + [1]
[2 0 2] [0]
[0 0 0] [0]
* Path {11}->{10}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
and(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]
gt(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]
s(x1) = [0 0 0] x1 + [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]
double(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
false() = [0]
[0]
[0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gt^#(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_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
and^#(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [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_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
double^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9() = [0]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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: {double^#(0()) -> c_9()}
Weak Rules: {double^#(s(x)) -> c_10(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [0]
[0 1 1] [1]
[0 0 0] [0]
0() = [2]
[2]
[2]
double^#(x1) = [2 2 2] x1 + [0]
[0 6 0] [0]
[0 0 2] [0]
c_9() = [1]
[0]
[0]
c_10(x1) = [1 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(true(), x, y) ->
c_0(f^#(and(gt(x, y), gt(y, s(s(0())))),
plus(s(0()), x),
double(y)))
, 2: gt^#(0(), v) -> c_1()
, 3: gt^#(s(u), 0()) -> c_2()
, 4: gt^#(s(u), s(v)) -> c_3(gt^#(u, v))
, 5: and^#(x, true()) -> c_4(x)
, 6: and^#(x, false()) -> c_5()
, 7: plus^#(n, 0()) -> c_6(n)
, 8: plus^#(n, s(m)) -> c_7(plus^#(n, m))
, 9: plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, 10: double^#(0()) -> c_9()
, 11: double^#(s(x)) -> c_10(double^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{8,9} [ inherited ]
|
`->{7} [ MAYBE ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^2)) ]
|
|->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {4}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(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: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_3(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(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: {gt^#(0(), v) -> c_1()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 1] [2]
0() = [2]
[2]
gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_1() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(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: {gt^#(s(u), 0()) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [2]
[0 1] [2]
0() = [0]
[2]
gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_2() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [1 1] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
[2]
and^#(x1, x2) = [7 7] x1 + [2 2] x2 + [7]
[7 7] [2 2] [3]
c_4(x1) = [1 3] x1 + [0]
[1 1] [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [2]
[2]
and^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_5() = [0]
[1]
* Path {8,9}: inherited
---------------------
This path is subsumed by the proof of path {8,9}->{7}.
* Path {8,9}->{7}: MAYBE
----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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:
{ plus^#(n, s(m)) -> c_7(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_6(n)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
* Path {11}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [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_10(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {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_10(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11}->{10}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
double(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
double^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [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_9()}
Weak Rules: {double^#(s(x)) -> c_10(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {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_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [5]
[2 0] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(true(), x, y) ->
c_0(f^#(and(gt(x, y), gt(y, s(s(0())))),
plus(s(0()), x),
double(y)))
, 2: gt^#(0(), v) -> c_1()
, 3: gt^#(s(u), 0()) -> c_2()
, 4: gt^#(s(u), s(v)) -> c_3(gt^#(u, v))
, 5: and^#(x, true()) -> c_4(x)
, 6: and^#(x, false()) -> c_5()
, 7: plus^#(n, 0()) -> c_6(n)
, 8: plus^#(n, s(m)) -> c_7(plus^#(n, m))
, 9: plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, 10: double^#(0()) -> c_9()
, 11: double^#(s(x)) -> c_10(double^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{11} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{8,9} [ inherited ]
|
`->{7} [ MAYBE ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)
, and(x, true()) -> x
, and(x, false()) -> false()
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(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: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(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: {gt^#(0(), v) -> c_1()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_1() = [1]
c_3(x1) = [1] x1 + [5]
* Path {4}->{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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(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: {gt^#(s(u), 0()) -> c_2()}
Weak Rules: {gt^#(s(u), s(v)) -> c_3(gt^#(u, v))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
0() = [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2() = [1]
c_3(x1) = [1] x1 + [7]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
and^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, true()) -> c_4(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [5]
and^#(x1, x2) = [7] x1 + [3] x2 + [0]
c_4(x1) = [1] x1 + [0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {and^#(x, false()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(and^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
false() = [7]
and^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_5() = [1]
* Path {8,9}: inherited
---------------------
This path is subsumed by the proof of path {8,9}->{7}.
* Path {8,9}->{7}: MAYBE
----------------------
The usable rules for this path are:
{ plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
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:
{ plus^#(n, s(m)) -> c_7(plus^#(n, m))
, plus^#(plus(n, m), u) -> c_8(plus^#(n, plus(m, u)))
, plus^#(n, 0()) -> c_6(n)
, plus(n, 0()) -> n
, plus(n, s(m)) -> s(plus(n, m))
, plus(plus(n, m), u) -> plus(n, plus(m, u))}
Proof Output:
The input cannot be shown compatible
* Path {11}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [3] x1 + [0]
c_9() = [0]
c_10(x1) = [1] 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: {double^#(s(x)) -> c_10(double^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
double^#(x1) = [2] x1 + [0]
c_10(x1) = [1] x1 + [7]
* Path {11}->{10}: 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(and) = {}, Uargs(gt) = {}, Uargs(s) = {},
Uargs(plus) = {}, Uargs(double) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(gt^#) = {}, Uargs(c_3) = {},
Uargs(and^#) = {}, Uargs(c_4) = {}, Uargs(plus^#) = {},
Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
and(x1, x2) = [0] x1 + [0] x2 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
double(x1) = [0] x1 + [0]
false() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
and^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
double^#(x1) = [0] x1 + [0]
c_9() = [0]
c_10(x1) = [1] 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: {double^#(0()) -> c_9()}
Weak Rules: {double^#(s(x)) -> c_10(double^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
double^#(x1) = [2] x1 + [0]
c_9() = [1]
c_10(x1) = [1] x1 + [0]
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.