Tool CaT
stdout:
MAYBE
Problem:
times(x,plus(y,s(z))) -> plus(times(x,plus(y,times(s(z),0()))),times(x,s(z)))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
plus(zero(),y) -> y
plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
id(x) -> x
if(true(),x,y) -> x
if(false(),x,y) -> y
not(x) -> if(x,false(),true())
gt(s(x),zero()) -> true()
gt(zero(),y) -> false()
gt(s(x),s(y)) -> gt(x,y)
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: times^#(x, plus(y, s(z))) ->
c_0(plus^#(times(x, plus(y, times(s(z), 0()))), times(x, s(z))))
, 2: times^#(x, 0()) -> c_1()
, 3: times^#(x, s(y)) -> c_2(plus^#(times(x, y), x))
, 4: plus^#(s(x), s(y)) ->
c_3(plus^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
, 5: plus^#(s(x), x) ->
c_4(plus^#(if(gt(x, x), id(x), id(x)), s(x)))
, 6: plus^#(zero(), y) -> c_5()
, 7: plus^#(id(x), s(y)) ->
c_6(plus^#(x, if(gt(s(y), y), y, s(y))))
, 8: id^#(x) -> c_7()
, 9: if^#(true(), x, y) -> c_8()
, 10: if^#(false(), x, y) -> c_9()
, 11: not^#(x) -> c_10(if^#(x, false(), true()))
, 12: gt^#(s(x), zero()) -> c_11()
, 13: gt^#(zero(), y) -> c_12()
, 14: gt^#(s(x), s(y)) -> c_13(gt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{14} [ YES(?,O(n^1)) ]
|
|->{12} [ YES(?,O(n^1)) ]
|
`->{13} [ YES(?,O(n^1)) ]
->{11} [ YES(?,O(1)) ]
|
|->{9} [ YES(?,O(1)) ]
|
`->{10} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{3} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7,5,4} [ inherited ]
|
`->{6} [ NA ]
->{2} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{6} [ NA ]
|
`->{7,5,4} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{7,5,4}->{6}.
* Path {1}->{6}: NA
-----------------
The usable rules for this path are:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {1}->{7,5,4}: inherited
----------------------------
This path is subsumed by the proof of path {1}->{7,5,4}->{6}.
* Path {1}->{7,5,4}->{6}: NA
--------------------------
The usable rules for this path are:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {2}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(id^#) = {}, Uargs(if^#) = {}, Uargs(not^#) = {},
Uargs(c_10) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9() = [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {times^#(x, 0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_1() = [1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{7,5,4}->{6}.
* Path {3}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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:
{ times^#(x, s(y)) -> c_2(plus^#(times(x, y), x))
, plus^#(zero(), y) -> c_5()
, times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {3}->{7,5,4}: inherited
----------------------------
This path is subsumed by the proof of path {3}->{7,5,4}->{6}.
* Path {3}->{7,5,4}->{6}: NA
--------------------------
The usable rules for this path are:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {8}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(id^#) = {}, Uargs(if^#) = {}, Uargs(not^#) = {},
Uargs(c_10) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9() = [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {id^#(x) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(id^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
id^#(x1) = [0] x1 + [7]
c_7() = [0]
* Path {11}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(id^#) = {}, Uargs(if^#) = {}, Uargs(not^#) = {},
Uargs(c_10) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7() = [0]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9() = [0]
not^#(x1) = [3] x1 + [0]
c_10(x1) = [3] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {not^#(x) -> c_10(if^#(x, false(), true()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [0]
false() = [0]
if^#(x1, x2, x3) = [7] x1 + [0] x2 + [0] x3 + [4]
not^#(x1) = [7] x1 + [7]
c_10(x1) = [0] x1 + [3]
* Path {11}->{9}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(id^#) = {}, Uargs(if^#) = {}, Uargs(not^#) = {},
Uargs(c_10) = {1}, Uargs(gt^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9() = [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {if^#(true(), x, y) -> c_8()}
Weak Rules: {not^#(x) -> c_10(if^#(x, false(), true()))}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
false() = [0]
if^#(x1, x2, x3) = [2] x1 + [0] x2 + [0] x3 + [0]
c_8() = [1]
not^#(x1) = [7] x1 + [7]
c_10(x1) = [2] x1 + [7]
* Path {11}->{10}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(id^#) = {}, Uargs(if^#) = {}, Uargs(not^#) = {},
Uargs(c_10) = {1}, Uargs(gt^#) = {}, Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9() = [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {if^#(false(), x, y) -> c_9()}
Weak Rules: {not^#(x) -> c_10(if^#(x, false(), true()))}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [0]
false() = [2]
if^#(x1, x2, x3) = [2] x1 + [0] x2 + [0] x3 + [4]
c_9() = [1]
not^#(x1) = [7] x1 + [7]
c_10(x1) = [1] x1 + [3]
* Path {14}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(id^#) = {}, Uargs(if^#) = {}, Uargs(not^#) = {},
Uargs(c_10) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9() = [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(x), s(y)) -> c_13(gt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_13(x1) = [1] x1 + [7]
* Path {14}->{12}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(id^#) = {}, Uargs(if^#) = {}, Uargs(not^#) = {},
Uargs(c_10) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9() = [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(s(x), zero()) -> c_11()}
Weak Rules: {gt^#(s(x), s(y)) -> c_13(gt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
zero() = [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_11() = [1]
c_13(x1) = [1] x1 + [7]
* Path {14}->{13}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
Uargs(id^#) = {}, Uargs(if^#) = {}, Uargs(not^#) = {},
Uargs(c_10) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7() = [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9() = [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gt^#(zero(), y) -> c_12()}
Weak Rules: {gt^#(s(x), s(y)) -> c_13(gt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
zero() = [2]
gt^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_12() = [1]
c_13(x1) = [1] x1 + [2]
2) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
3) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
4) '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:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: times^#(x, plus(y, s(z))) ->
c_0(plus^#(times(x, plus(y, times(s(z), 0()))), times(x, s(z))))
, 2: times^#(x, 0()) -> c_1()
, 3: times^#(x, s(y)) -> c_2(plus^#(times(x, y), x))
, 4: plus^#(s(x), s(y)) ->
c_3(plus^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
, 5: plus^#(s(x), x) ->
c_4(plus^#(if(gt(x, x), id(x), id(x)), s(x)))
, 6: plus^#(zero(), y) -> c_5(y)
, 7: plus^#(id(x), s(y)) ->
c_6(plus^#(x, if(gt(s(y), y), y, s(y))))
, 8: id^#(x) -> c_7(x)
, 9: if^#(true(), x, y) -> c_8(x)
, 10: if^#(false(), x, y) -> c_9(y)
, 11: not^#(x) -> c_10(if^#(x, false(), true()))
, 12: gt^#(s(x), zero()) -> c_11()
, 13: gt^#(zero(), y) -> c_12()
, 14: gt^#(s(x), s(y)) -> c_13(gt^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{14} [ YES(?,O(n^1)) ]
|
|->{12} [ YES(?,O(n^1)) ]
|
`->{13} [ YES(?,O(n^1)) ]
->{11} [ YES(?,O(1)) ]
|
|->{9} [ YES(?,O(1)) ]
|
`->{10} [ YES(?,O(1)) ]
->{8} [ YES(?,O(1)) ]
->{3} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7,5,4} [ inherited ]
|
`->{6} [ NA ]
->{2} [ YES(?,O(1)) ]
->{1} [ inherited ]
|
|->{6} [ NA ]
|
`->{7,5,4} [ inherited ]
|
`->{6} [ NA ]
Sub-problems:
-------------
* Path {1}: inherited
-------------------
This path is subsumed by the proof of path {1}->{7,5,4}->{6}.
* Path {1}->{6}: NA
-----------------
The usable rules for this path are:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {1}->{7,5,4}: inherited
----------------------------
This path is subsumed by the proof of path {1}->{7,5,4}->{6}.
* Path {1}->{7,5,4}->{6}: NA
--------------------------
The usable rules for this path are:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {2}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(id^#) = {}, Uargs(c_7) = {},
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(not^#) = {}, Uargs(c_10) = {}, Uargs(gt^#) = {},
Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {times^#(x, 0()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(times^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
times^#(x1, x2) = [0] x1 + [1] x2 + [7]
c_1() = [1]
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{7,5,4}->{6}.
* Path {3}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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:
{ times^#(x, s(y)) -> c_2(plus^#(times(x, y), x))
, plus^#(zero(), y) -> c_5(y)
, times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {3}->{7,5,4}: inherited
----------------------------
This path is subsumed by the proof of path {3}->{7,5,4}->{6}.
* Path {3}->{7,5,4}->{6}: NA
--------------------------
The usable rules for this path are:
{ times(x, plus(y, s(z))) ->
plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))
, times(x, 0()) -> 0()
, times(x, s(y)) -> plus(times(x, y), x)
, plus(s(x), s(y)) ->
s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
, plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
, plus(zero(), y) -> y
, plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
, id(x) -> x
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, not(x) -> if(x, false(), true())
, gt(s(x), zero()) -> true()
, gt(zero(), y) -> false()
, gt(s(x), s(y)) -> gt(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 {8}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(id^#) = {}, Uargs(c_7) = {},
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(not^#) = {}, Uargs(c_10) = {}, Uargs(gt^#) = {},
Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [3] x1 + [0]
c_7(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {id^#(x) -> c_7(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(id^#) = {}, Uargs(c_7) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
id^#(x1) = [7] x1 + [7]
c_7(x1) = [1] x1 + [0]
* Path {11}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(id^#) = {}, Uargs(c_7) = {},
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(not^#) = {}, Uargs(c_10) = {}, Uargs(gt^#) = {},
Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
not^#(x1) = [3] x1 + [0]
c_10(x1) = [3] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {not^#(x) -> c_10(if^#(x, false(), true()))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(not^#) = {}, Uargs(c_10) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [0]
false() = [0]
if^#(x1, x2, x3) = [7] x1 + [0] x2 + [0] x3 + [4]
not^#(x1) = [7] x1 + [7]
c_10(x1) = [0] x1 + [3]
* Path {11}->{9}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(id^#) = {}, Uargs(c_7) = {},
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(not^#) = {}, Uargs(c_10) = {1}, Uargs(gt^#) = {},
Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {if^#(true(), x, y) -> c_8(x)}
Weak Rules: {not^#(x) -> c_10(if^#(x, false(), true()))}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(not^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [2]
false() = [0]
if^#(x1, x2, x3) = [2] x1 + [2] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [1]
not^#(x1) = [7] x1 + [7]
c_10(x1) = [2] x1 + [7]
* Path {11}->{10}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(id^#) = {}, Uargs(c_7) = {},
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(not^#) = {}, Uargs(c_10) = {1}, Uargs(gt^#) = {},
Uargs(c_13) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [1] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [1] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(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: {if^#(false(), x, y) -> c_9(y)}
Weak Rules: {not^#(x) -> c_10(if^#(x, false(), true()))}
Proof Output:
The following argument positions are usable:
Uargs(if^#) = {}, Uargs(c_9) = {}, Uargs(not^#) = {},
Uargs(c_10) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
true() = [0]
false() = [2]
if^#(x1, x2, x3) = [2] x1 + [0] x2 + [0] x3 + [4]
c_9(x1) = [0] x1 + [1]
not^#(x1) = [7] x1 + [7]
c_10(x1) = [1] x1 + [3]
* Path {14}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(id^#) = {}, Uargs(c_7) = {},
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(not^#) = {}, Uargs(c_10) = {}, Uargs(gt^#) = {},
Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(x), s(y)) -> c_13(gt^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_13(x1) = [1] x1 + [7]
* Path {14}->{12}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(id^#) = {}, Uargs(c_7) = {},
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(not^#) = {}, Uargs(c_10) = {}, Uargs(gt^#) = {},
Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(s(x), zero()) -> c_11()}
Weak Rules: {gt^#(s(x), s(y)) -> c_13(gt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
zero() = [2]
gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_11() = [1]
c_13(x1) = [1] x1 + [7]
* Path {14}->{13}: 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(times) = {}, Uargs(plus) = {}, Uargs(s) = {}, Uargs(if) = {},
Uargs(gt) = {}, Uargs(not) = {}, Uargs(id) = {},
Uargs(times^#) = {}, Uargs(c_0) = {}, Uargs(plus^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(c_6) = {}, Uargs(id^#) = {}, Uargs(c_7) = {},
Uargs(if^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
Uargs(not^#) = {}, Uargs(c_10) = {}, Uargs(gt^#) = {},
Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
times(x1, x2) = [0] x1 + [0] x2 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
gt(x1, x2) = [0] x1 + [0] x2 + [0]
not(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
zero() = [0]
true() = [0]
false() = [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_10(x1) = [0] x1 + [0]
gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_11() = [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gt^#(zero(), y) -> c_12()}
Weak Rules: {gt^#(s(x), s(y)) -> c_13(gt^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_13) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
zero() = [2]
gt^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_12() = [1]
c_13(x1) = [1] x1 + [2]
2) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
3) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
4) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.