Tool CaT
stdout:
MAYBE
Problem:
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> times(x,exp(x,y))
ge(x,0()) -> true()
ge(0(),s(x)) -> false()
ge(s(x),s(y)) -> ge(x,y)
tower(x,y) -> towerIter(0(),x,y,s(0()))
towerIter(c,x,y,z) -> help(ge(c,x),c,x,y,z)
help(true(),c,x,y,z) -> z
help(false(),c,x,y,z) -> towerIter(s(c),x,y,exp(y,z))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, ge(x, 0()) -> true()
, ge(0(), s(x)) -> false()
, ge(s(x), s(y)) -> ge(x, y)
, tower(x, y) -> towerIter(0(), x, y, s(0()))
, towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
, help(true(), c, x, y, z) -> z
, help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(0(), x) -> c_0()
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: times^#(0(), y) -> c_2()
, 4: times^#(s(x), y) -> c_3(plus^#(y, times(x, y)))
, 5: exp^#(x, 0()) -> c_4()
, 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, 7: ge^#(x, 0()) -> c_6()
, 8: ge^#(0(), s(x)) -> c_7()
, 9: ge^#(s(x), s(y)) -> c_8(ge^#(x, y))
, 10: tower^#(x, y) -> c_9(towerIter^#(0(), x, y, s(0())))
, 11: towerIter^#(c, x, y, z) -> c_10(help^#(ge(c, x), c, x, y, z))
, 12: help^#(true(), c, x, y, z) -> c_11()
, 13: help^#(false(), c, x, y, z) ->
c_12(towerIter^#(s(c), x, y, exp(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ YES(?,O(n^1)) ]
|
|->{7} [ YES(?,O(n^3)) ]
|
`->{8} [ YES(?,O(n^2)) ]
->{6} [ inherited ]
|
|->{3} [ MAYBE ]
|
`->{4} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(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]
exp(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]
ge(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
tower(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]
towerIter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
exp^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(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() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tower^#(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(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: {exp^#(x, 0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(exp^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
exp^#(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}: inherited
-------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, times^#(0(), y) -> c_2()
, exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{4}: inherited
------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{1}: NA
----------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{4}->{2}: inherited
-----------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{2}->{1}: NA
---------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {9}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(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]
exp(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]
ge(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
tower(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]
towerIter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
exp^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tower^#(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(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: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
ge^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_8(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {9}->{7}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(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]
exp(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]
ge(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
tower(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]
towerIter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
exp^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(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() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tower^#(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {ge^#(x, 0()) -> c_6()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
ge^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_6() = [1]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {9}->{8}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(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]
exp(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]
ge(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
tower(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]
towerIter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
exp^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(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() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tower^#(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(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: {ge^#(0(), s(x)) -> c_7()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
ge^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_7() = [1]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, ge(x, 0()) -> true()
, ge(0(), s(x)) -> false()
, ge(s(x), s(y)) -> ge(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(0(), x) -> c_0()
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: times^#(0(), y) -> c_2()
, 4: times^#(s(x), y) -> c_3(plus^#(y, times(x, y)))
, 5: exp^#(x, 0()) -> c_4()
, 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, 7: ge^#(x, 0()) -> c_6()
, 8: ge^#(0(), s(x)) -> c_7()
, 9: ge^#(s(x), s(y)) -> c_8(ge^#(x, y))
, 10: tower^#(x, y) -> c_9(towerIter^#(0(), x, y, s(0())))
, 11: towerIter^#(c, x, y, z) -> c_10(help^#(ge(c, x), c, x, y, z))
, 12: help^#(true(), c, x, y, z) -> c_11()
, 13: help^#(false(), c, x, y, z) ->
c_12(towerIter^#(s(c), x, y, exp(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ YES(?,O(n^2)) ]
|
|->{7} [ YES(?,O(n^2)) ]
|
`->{8} [ YES(?,O(n^2)) ]
->{6} [ inherited ]
|
|->{3} [ MAYBE ]
|
`->{4} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
towerIter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
exp^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(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: {exp^#(x, 0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(exp^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
exp^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_4() = [0]
[1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, times^#(0(), y) -> c_2()
, exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{4}: inherited
------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{1}: NA
----------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{4}->{2}: inherited
-----------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{2}->{1}: NA
---------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {9}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
towerIter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
exp^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(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: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
ge^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_8(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {9}->{7}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
towerIter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
exp^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(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: {ge^#(x, 0()) -> c_6()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
ge^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_6() = [1]
[0]
c_8(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {9}->{8}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
towerIter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
exp^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(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: {ge^#(0(), s(x)) -> c_7()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
ge^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_7() = [1]
[0]
c_8(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, ge(x, 0()) -> true()
, ge(0(), s(x)) -> false()
, ge(s(x), s(y)) -> ge(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(0(), x) -> c_0()
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: times^#(0(), y) -> c_2()
, 4: times^#(s(x), y) -> c_3(plus^#(y, times(x, y)))
, 5: exp^#(x, 0()) -> c_4()
, 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, 7: ge^#(x, 0()) -> c_6()
, 8: ge^#(0(), s(x)) -> c_7()
, 9: ge^#(s(x), s(y)) -> c_8(ge^#(x, y))
, 10: tower^#(x, y) -> c_9(towerIter^#(0(), x, y, s(0())))
, 11: towerIter^#(c, x, y, z) -> c_10(help^#(ge(c, x), c, x, y, z))
, 12: help^#(true(), c, x, y, z) -> c_11()
, 13: help^#(false(), c, x, y, z) ->
c_12(towerIter^#(s(c), x, y, exp(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ YES(?,O(n^1)) ]
|
|->{7} [ YES(?,O(n^1)) ]
|
`->{8} [ YES(?,O(n^1)) ]
->{6} [ inherited ]
|
|->{3} [ MAYBE ]
|
`->{4} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
exp(x1, x2) = [0] x1 + [0] x2 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
tower(x1, x2) = [0] x1 + [0] x2 + [0]
towerIter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
help(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
towerIter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_10(x1) = [0] x1 + [0]
help^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_11() = [0]
c_12(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: {exp^#(x, 0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(exp^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
exp^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_4() = [1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, times^#(0(), y) -> c_2()
, exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{4}: inherited
------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{1}: NA
----------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{4}->{2}: inherited
-----------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{2}->{1}: NA
---------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {9}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
exp(x1, x2) = [0] x1 + [0] x2 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
tower(x1, x2) = [0] x1 + [0] x2 + [0]
towerIter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
help(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
ge^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
towerIter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_10(x1) = [0] x1 + [0]
help^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_11() = [0]
c_12(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: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_8(x1) = [1] x1 + [7]
* Path {9}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
exp(x1, x2) = [0] x1 + [0] x2 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
tower(x1, x2) = [0] x1 + [0] x2 + [0]
towerIter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
help(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
towerIter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_10(x1) = [0] x1 + [0]
help^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_11() = [0]
c_12(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: {ge^#(x, 0()) -> c_6()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_6() = [1]
c_8(x1) = [1] x1 + [7]
* Path {9}->{8}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_1) = {}, Uargs(times^#) = {}, Uargs(c_3) = {},
Uargs(exp^#) = {}, Uargs(c_5) = {}, Uargs(ge^#) = {},
Uargs(c_8) = {1}, Uargs(tower^#) = {}, Uargs(c_9) = {},
Uargs(towerIter^#) = {}, Uargs(c_10) = {}, Uargs(help^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
exp(x1, x2) = [0] x1 + [0] x2 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
tower(x1, x2) = [0] x1 + [0] x2 + [0]
towerIter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
help(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
towerIter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_10(x1) = [0] x1 + [0]
help^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_11() = [0]
c_12(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: {ge^#(0(), s(x)) -> c_7()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7() = [1]
c_8(x1) = [1] x1 + [7]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, ge(x, 0()) -> true()
, ge(0(), s(x)) -> false()
, ge(s(x), s(y)) -> ge(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, ge(x, 0()) -> true()
, ge(0(), s(x)) -> false()
, ge(s(x), s(y)) -> ge(x, y)
, tower(x, y) -> towerIter(0(), x, y, s(0()))
, towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
, help(true(), c, x, y, z) -> z
, help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(0(), x) -> c_0(x)
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: times^#(0(), y) -> c_2()
, 4: times^#(s(x), y) -> c_3(plus^#(y, times(x, y)))
, 5: exp^#(x, 0()) -> c_4()
, 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, 7: ge^#(x, 0()) -> c_6()
, 8: ge^#(0(), s(x)) -> c_7()
, 9: ge^#(s(x), s(y)) -> c_8(ge^#(x, y))
, 10: tower^#(x, y) -> c_9(towerIter^#(0(), x, y, s(0())))
, 11: towerIter^#(c, x, y, z) -> c_10(help^#(ge(c, x), c, x, y, z))
, 12: help^#(true(), c, x, y, z) -> c_11(z)
, 13: help^#(false(), c, x, y, z) ->
c_12(towerIter^#(s(c), x, y, exp(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ YES(?,O(n^1)) ]
|
|->{7} [ YES(?,O(n^3)) ]
|
`->{8} [ YES(?,O(n^2)) ]
->{6} [ inherited ]
|
|->{3} [ MAYBE ]
|
`->{4} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(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]
exp(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]
ge(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
tower(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]
towerIter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
exp^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(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() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tower^#(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(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: {exp^#(x, 0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(exp^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
exp^#(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}: inherited
-------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, times^#(0(), y) -> c_2()
, exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{4}: inherited
------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{1}: NA
----------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{4}->{2}: inherited
-----------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{2}->{1}: NA
---------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {9}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(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]
exp(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]
ge(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
tower(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]
towerIter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
exp^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(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_6() = [0]
[0]
[0]
c_7() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tower^#(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(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: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
ge^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_8(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {9}->{7}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(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]
exp(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]
ge(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
tower(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]
towerIter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
exp^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(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() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tower^#(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {ge^#(x, 0()) -> c_6()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
ge^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_6() = [1]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {9}->{8}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times(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]
exp(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]
ge(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]
true() = [0]
[0]
[0]
false() = [0]
[0]
[0]
tower(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]
towerIter(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
plus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
times^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
exp^#(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(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
ge^#(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() = [0]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
tower^#(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_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0 0 0] x5 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(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: {ge^#(0(), s(x)) -> c_7()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
[0]
s(x1) = [1 1 2] x1 + [0]
[0 1 0] [2]
[0 0 0] [0]
ge^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
[7 1 0] [4 0 0] [0]
[4 2 0] [4 0 0] [0]
c_7() = [1]
[0]
[0]
c_8(x1) = [1 0 0] x1 + [3]
[2 0 0] [2]
[0 0 0] [2]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, ge(x, 0()) -> true()
, ge(0(), s(x)) -> false()
, ge(s(x), s(y)) -> ge(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(0(), x) -> c_0(x)
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: times^#(0(), y) -> c_2()
, 4: times^#(s(x), y) -> c_3(plus^#(y, times(x, y)))
, 5: exp^#(x, 0()) -> c_4()
, 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, 7: ge^#(x, 0()) -> c_6()
, 8: ge^#(0(), s(x)) -> c_7()
, 9: ge^#(s(x), s(y)) -> c_8(ge^#(x, y))
, 10: tower^#(x, y) -> c_9(towerIter^#(0(), x, y, s(0())))
, 11: towerIter^#(c, x, y, z) -> c_10(help^#(ge(c, x), c, x, y, z))
, 12: help^#(true(), c, x, y, z) -> c_11(z)
, 13: help^#(false(), c, x, y, z) ->
c_12(towerIter^#(s(c), x, y, exp(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ YES(?,O(n^2)) ]
|
|->{7} [ YES(?,O(n^2)) ]
|
`->{8} [ YES(?,O(n^2)) ]
->{6} [ inherited ]
|
|->{3} [ MAYBE ]
|
`->{4} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
towerIter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
exp^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {exp^#(x, 0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(exp^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
exp^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
[0 0] [2 2] [7]
c_4() = [0]
[1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, times^#(0(), y) -> c_2()
, exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{4}: inherited
------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{1}: NA
----------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{4}->{2}: inherited
-----------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{2}->{1}: NA
---------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {9}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
towerIter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
exp^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
ge^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_8(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {9}->{7}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
towerIter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
exp^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {ge^#(x, 0()) -> c_6()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
ge^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_6() = [1]
[0]
c_8(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {9}->{8}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
ge(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
towerIter(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
help(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
exp^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
ge^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
towerIter^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
help^#(x1, x2, x3, x4, x5) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(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: {ge^#(0(), s(x)) -> c_7()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 6] x1 + [2]
[0 1] [2]
ge^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
[1 2] [2 0] [0]
c_7() = [1]
[0]
c_8(x1) = [1 0] x1 + [7]
[0 0] [7]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, ge(x, 0()) -> true()
, ge(0(), s(x)) -> false()
, ge(s(x), s(y)) -> ge(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(0(), x) -> c_0(x)
, 2: plus^#(s(x), y) -> c_1(plus^#(x, y))
, 3: times^#(0(), y) -> c_2()
, 4: times^#(s(x), y) -> c_3(plus^#(y, times(x, y)))
, 5: exp^#(x, 0()) -> c_4()
, 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, 7: ge^#(x, 0()) -> c_6()
, 8: ge^#(0(), s(x)) -> c_7()
, 9: ge^#(s(x), s(y)) -> c_8(ge^#(x, y))
, 10: tower^#(x, y) -> c_9(towerIter^#(0(), x, y, s(0())))
, 11: towerIter^#(c, x, y, z) -> c_10(help^#(ge(c, x), c, x, y, z))
, 12: help^#(true(), c, x, y, z) -> c_11(z)
, 13: help^#(false(), c, x, y, z) ->
c_12(towerIter^#(s(c), x, y, exp(y, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ YES(?,O(n^1)) ]
|
|->{7} [ YES(?,O(n^1)) ]
|
`->{8} [ YES(?,O(n^1)) ]
->{6} [ inherited ]
|
|->{3} [ MAYBE ]
|
`->{4} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ inherited ]
|
`->{1} [ NA ]
->{5} [ YES(?,O(1)) ]
Sub-problems:
-------------
* 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
exp(x1, x2) = [0] x1 + [0] x2 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
tower(x1, x2) = [0] x1 + [0] x2 + [0]
towerIter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
help(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
towerIter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_10(x1) = [0] x1 + [0]
help^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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: {exp^#(x, 0()) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(exp^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
exp^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_4() = [1]
* Path {6}: inherited
-------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{3}: MAYBE
--------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
, times^#(0(), y) -> c_2()
, exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {6}->{4}: inherited
------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{1}: NA
----------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {6}->{4}->{2}: inherited
-----------------------------
This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
* Path {6}->{4}->{2}->{1}: NA
---------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {9}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
exp(x1, x2) = [0] x1 + [0] x2 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
tower(x1, x2) = [0] x1 + [0] x2 + [0]
towerIter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
help(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
ge^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
towerIter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_10(x1) = [0] x1 + [0]
help^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_8(x1) = [1] x1 + [7]
* Path {9}->{7}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
exp(x1, x2) = [0] x1 + [0] x2 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
tower(x1, x2) = [0] x1 + [0] x2 + [0]
towerIter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
help(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
towerIter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_10(x1) = [0] x1 + [0]
help^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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: {ge^#(x, 0()) -> c_6()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_6() = [1]
c_8(x1) = [1] x1 + [7]
* Path {9}->{8}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
Uargs(exp) = {}, Uargs(ge) = {}, Uargs(tower) = {},
Uargs(towerIter) = {}, Uargs(help) = {}, Uargs(plus^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
Uargs(ge^#) = {}, Uargs(c_8) = {1}, Uargs(tower^#) = {},
Uargs(c_9) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {},
Uargs(help^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
exp(x1, x2) = [0] x1 + [0] x2 + [0]
ge(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
tower(x1, x2) = [0] x1 + [0] x2 + [0]
towerIter(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
help(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
ge^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6() = [0]
c_7() = [0]
c_8(x1) = [1] x1 + [0]
tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
towerIter^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_10(x1) = [0] x1 + [0]
help^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
c_11(x1) = [0] x1 + [0]
c_12(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: {ge^#(0(), s(x)) -> c_7()}
Weak Rules: {ge^#(s(x), s(y)) -> c_8(ge^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(ge^#) = {}, Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
ge^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_7() = [1]
c_8(x1) = [1] x1 + [7]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ exp(x, 0()) -> s(0())
, exp(x, s(y)) -> times(x, exp(x, y))
, ge(x, 0()) -> true()
, ge(0(), s(x)) -> false()
, ge(s(x), s(y)) -> ge(x, y)
, times(0(), y) -> 0()
, times(s(x), y) -> plus(y, times(x, y))
, plus(0(), x) -> x
, plus(s(x), y) -> s(plus(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.