Tool CaT
stdout:
MAYBE
Problem:
check(0()) -> zero()
check(s(0())) -> odd()
check(s(s(0()))) -> even()
check(s(s(s(x)))) -> check(s(x))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(x,y) -> timesIter(x,y,0())
timesIter(x,y,z) -> if(check(x),x,y,z,plus(z,y))
p(s(x)) -> x
p(0()) -> 0()
if(zero(),x,y,z,u) -> z
if(odd(),x,y,z,u) -> timesIter(p(x),y,u)
if(even(),x,y,z,u) -> plus(timesIter(half(x),y,half(z)),timesIter(half(x),y,half(s(z))))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
TIMEOUT
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: TIMEOUT
Input Problem: innermost runtime-complexity with respect to
Rules:
{ check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, times(x, y) -> timesIter(x, y, 0())
, timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
, p(s(x)) -> x
, p(0()) -> 0()
, if(zero(), x, y, z, u) -> z
, if(odd(), x, y, z, u) -> timesIter(p(x), y, u)
, if(even(), x, y, z, u) ->
plus(timesIter(half(x), y, half(z)),
timesIter(half(x), y, half(s(z))))}
Proof Output:
Computation stopped due to timeout after 60.0 secondsTool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, times(x, y) -> timesIter(x, y, 0())
, timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
, p(s(x)) -> x
, p(0()) -> 0()
, if(zero(), x, y, z, u) -> z
, if(odd(), x, y, z, u) -> timesIter(p(x), y, u)
, if(even(), x, y, z, u) ->
plus(timesIter(half(x), y, half(z)),
timesIter(half(x), y, half(s(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: check^#(0()) -> c_0()
, 2: check^#(s(0())) -> c_1()
, 3: check^#(s(s(0()))) -> c_2()
, 4: check^#(s(s(s(x)))) -> c_3(check^#(s(x)))
, 5: half^#(0()) -> c_4()
, 6: half^#(s(0())) -> c_5()
, 7: half^#(s(s(x))) -> c_6(half^#(x))
, 8: plus^#(0(), y) -> c_7(y)
, 9: plus^#(s(x), y) -> c_8(plus^#(x, y))
, 10: times^#(x, y) -> c_9(timesIter^#(x, y, 0()))
, 11: timesIter^#(x, y, z) ->
c_10(if^#(check(x), x, y, z, plus(z, y)))
, 12: p^#(s(x)) -> c_11(x)
, 13: p^#(0()) -> c_12()
, 14: if^#(zero(), x, y, z, u) -> c_13(z)
, 15: if^#(odd(), x, y, z, u) -> c_14(timesIter^#(p(x), y, u))
, 16: if^#(even(), x, y, z, u) ->
c_15(plus^#(timesIter(half(x), y, half(z)),
timesIter(half(x), y, half(s(z)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ YES(?,O(1)) ]
->{12} [ YES(?,O(n^2)) ]
->{10} [ inherited ]
|
`->{11,15} [ inherited ]
|
|->{14} [ MAYBE ]
|
`->{16} [ inherited ]
|
|->{8} [ NA ]
|
`->{9} [ inherited ]
|
`->{8} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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: {check^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(check^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
check^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {1}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {check^#(s(s(s(x)))) -> c_3(check^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(check^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0 0] x1 + [0]
[0 1] [2]
check^#(x1) = [0 1] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {4}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {1}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {check^#(s(0())) -> c_1()}
Weak Rules: {check^#(s(s(s(x)))) -> c_3(check^#(s(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(check^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [0 1] x1 + [2]
[0 1] [1]
check^#(x1) = [2 0] x1 + [0]
[2 0] [0]
c_1() = [1]
[0]
c_3(x1) = [1 0] x1 + [2]
[0 0] [7]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {1}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {check^#(s(s(0()))) -> c_2()}
Weak Rules: {check^#(s(s(s(x)))) -> c_3(check^#(s(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(check^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [0]
[0 0] [2]
check^#(x1) = [1 0] x1 + [0]
[0 2] [0]
c_2() = [1]
[0]
c_3(x1) = [1 2] x1 + [0]
[0 0] [3]
* Path {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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
half^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {half^#(s(s(x))) -> c_6(half^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
half^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_6(x1) = [1 0] x1 + [5]
[2 0] [7]
* Path {7}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {half^#(0()) -> c_4()}
Weak Rules: {half^#(s(s(x))) -> c_6(half^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [1]
[0 0] [0]
half^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_4() = [1]
[0]
c_6(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {7}->{6}: 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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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: {half^#(s(0())) -> c_5()}
Weak Rules: {half^#(s(s(x))) -> c_6(half^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [1]
[0]
s(x1) = [1 1] x1 + [2]
[0 1] [1]
half^#(x1) = [2 1] x1 + [1]
[0 0] [7]
c_5() = [1]
[1]
c_6(x1) = [1 1] x1 + [3]
[0 0] [3]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,15}->{16}->{9}->{8}.
* Path {10}->{11,15}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,15}->{16}->{9}->{8}.
* Path {10}->{11,15}->{14}: MAYBE
-------------------------------
The usable rules for this path are:
{ check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, p(s(x)) -> x
, p(0()) -> 0()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {1}, Uargs(s) = {1}, Uargs(half) = {},
Uargs(plus) = {1}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {1}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {1}, Uargs(timesIter^#) = {1, 3},
Uargs(c_10) = {1}, Uargs(if^#) = {1, 2, 4, 5}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {1}, Uargs(c_14) = {1},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [2 0] x1 + [1]
[0 0] [0]
0() = [0]
[1]
zero() = [0]
[0]
s(x1) = [1 0] x1 + [2]
[0 1] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [2 0] x1 + [3 2] x2 + [2]
[0 0] [3 3] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [1 1] x1 + [0]
[0 1] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
timesIter^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [3 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3, x4, x5) = [3 0] x1 + [3 3] x2 + [0 0] x3 + [3 0] x4 + [3 0] x5 + [0]
[0 0] [0 0] [0 0] [0 0] [0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [1 0] x1 + [0]
[0 1] [0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(zero(), x, y, z, u) -> c_13(z)}
Weak Rules:
{ timesIter^#(x, y, z) ->
c_10(if^#(check(x), x, y, z, plus(z, y)))
, if^#(odd(), x, y, z, u) -> c_14(timesIter^#(p(x), y, u))
, times^#(x, y) -> c_9(timesIter^#(x, y, 0()))
, check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, p(s(x)) -> x
, p(0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* Path {10}->{11,15}->{16}: inherited
-----------------------------------
This path is subsumed by the proof of path {10}->{11,15}->{16}->{9}->{8}.
* Path {10}->{11,15}->{16}->{8}: NA
---------------------------------
The usable rules for this path are:
{ check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, p(s(x)) -> x
, p(0()) -> 0()
, half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
, if(zero(), x, y, z, u) -> z
, if(odd(), x, y, z, u) -> timesIter(p(x), y, u)
, if(even(), x, y, z, u) ->
plus(timesIter(half(x), y, half(z)),
timesIter(half(x), y, half(s(z))))}
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 {10}->{11,15}->{16}->{9}: inherited
----------------------------------------
This path is subsumed by the proof of path {10}->{11,15}->{16}->{9}->{8}.
* Path {10}->{11,15}->{16}->{9}->{8}: NA
--------------------------------------
The usable rules for this path are:
{ check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, p(s(x)) -> x
, p(0()) -> 0()
, half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
, if(zero(), x, y, z, u) -> z
, if(odd(), x, y, z, u) -> timesIter(p(x), y, u)
, if(even(), x, y, z, u) ->
plus(timesIter(half(x), y, half(z)),
timesIter(half(x), y, half(s(z))))}
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 {12}: 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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 1] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_11(x1) = [1 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(x)) -> c_11(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [2]
[0 0] [2]
p^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_11(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {13}: 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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
zero() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
odd() = [0]
[0]
even() = [0]
[0]
half(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
timesIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
if(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]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
check^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
half^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
timesIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(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]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12() = [0]
[0]
c_13(x1) = [0 0] x1 + [0]
[0 0] [0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
c_15(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: {p^#(0()) -> c_12()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_12() = [0]
[1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: check^#(0()) -> c_0()
, 2: check^#(s(0())) -> c_1()
, 3: check^#(s(s(0()))) -> c_2()
, 4: check^#(s(s(s(x)))) -> c_3(check^#(s(x)))
, 5: half^#(0()) -> c_4()
, 6: half^#(s(0())) -> c_5()
, 7: half^#(s(s(x))) -> c_6(half^#(x))
, 8: plus^#(0(), y) -> c_7(y)
, 9: plus^#(s(x), y) -> c_8(plus^#(x, y))
, 10: times^#(x, y) -> c_9(timesIter^#(x, y, 0()))
, 11: timesIter^#(x, y, z) ->
c_10(if^#(check(x), x, y, z, plus(z, y)))
, 12: p^#(s(x)) -> c_11(x)
, 13: p^#(0()) -> c_12()
, 14: if^#(zero(), x, y, z, u) -> c_13(z)
, 15: if^#(odd(), x, y, z, u) -> c_14(timesIter^#(p(x), y, u))
, 16: if^#(even(), x, y, z, u) ->
c_15(plus^#(timesIter(half(x), y, half(z)),
timesIter(half(x), y, half(s(z)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{13} [ YES(?,O(1)) ]
->{12} [ YES(?,O(n^1)) ]
->{10} [ inherited ]
|
`->{11,15} [ inherited ]
|
|->{14} [ MAYBE ]
|
`->{16} [ inherited ]
|
|->{8} [ NA ]
|
`->{9} [ inherited ]
|
`->{8} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(1)) ]
->{4} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(1)) ]
|
`->{3} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: 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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [0] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {check^#(0()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(check^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
check^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {1}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [1] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [3] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {check^#(s(s(s(x)))) -> c_3(check^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(check^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
check^#(x1) = [2] x1 + [0]
c_3(x1) = [1] x1 + [5]
* Path {4}->{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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {1}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [0] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {check^#(s(0())) -> c_1()}
Weak Rules: {check^#(s(s(s(x)))) -> c_3(check^#(s(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(check^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
check^#(x1) = [2] x1 + [0]
c_1() = [1]
c_3(x1) = [1] x1 + [0]
* Path {4}->{3}: 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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {1}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [0] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {check^#(s(s(0()))) -> c_2()}
Weak Rules: {check^#(s(s(s(x)))) -> c_3(check^#(s(x)))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(check^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [2]
check^#(x1) = [2] x1 + [4]
c_2() = [1]
c_3(x1) = [1] x1 + [0]
* Path {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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [1] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
half^#(x1) = [3] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {half^#(s(s(x))) -> c_6(half^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
half^#(x1) = [2] x1 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [0] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {half^#(0()) -> c_4()}
Weak Rules: {half^#(s(s(x))) -> c_6(half^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
half^#(x1) = [2] x1 + [4]
c_4() = [1]
c_6(x1) = [1] x1 + [0]
* Path {7}->{6}: YES(?,O(1))
--------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [0] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {half^#(s(0())) -> c_5()}
Weak Rules: {half^#(s(s(x))) -> c_6(half^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [1]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,15}->{16}->{9}->{8}.
* Path {10}->{11,15}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,15}->{16}->{9}->{8}.
* Path {10}->{11,15}->{14}: MAYBE
-------------------------------
The usable rules for this path are:
{ check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, p(s(x)) -> x
, p(0()) -> 0()}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(check) = {1}, Uargs(s) = {1}, Uargs(half) = {},
Uargs(plus) = {1}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {1}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {1}, Uargs(timesIter^#) = {1, 3},
Uargs(c_10) = {1}, Uargs(if^#) = {1, 2, 4, 5}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {1}, Uargs(c_14) = {1},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [3] x1 + [0]
0() = [2]
zero() = [1]
s(x1) = [1] x1 + [2]
odd() = [1]
even() = [1]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [2] x1 + [3] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [3] x1 + [3]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [1] x1 + [0]
timesIter^#(x1, x2, x3) = [3] x1 + [0] x2 + [3] x3 + [0]
c_10(x1) = [1] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [3] x1 + [3] x2 + [0] x3 + [3] x4 + [3] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [1] x1 + [0]
c_14(x1) = [1] x1 + [0]
c_15(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {if^#(zero(), x, y, z, u) -> c_13(z)}
Weak Rules:
{ timesIter^#(x, y, z) ->
c_10(if^#(check(x), x, y, z, plus(z, y)))
, if^#(odd(), x, y, z, u) -> c_14(timesIter^#(p(x), y, u))
, times^#(x, y) -> c_9(timesIter^#(x, y, 0()))
, check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, p(s(x)) -> x
, p(0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* Path {10}->{11,15}->{16}: inherited
-----------------------------------
This path is subsumed by the proof of path {10}->{11,15}->{16}->{9}->{8}.
* Path {10}->{11,15}->{16}->{8}: NA
---------------------------------
The usable rules for this path are:
{ check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, p(s(x)) -> x
, p(0()) -> 0()
, half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
, if(zero(), x, y, z, u) -> z
, if(odd(), x, y, z, u) -> timesIter(p(x), y, u)
, if(even(), x, y, z, u) ->
plus(timesIter(half(x), y, half(z)),
timesIter(half(x), y, half(s(z))))}
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 {10}->{11,15}->{16}->{9}: inherited
----------------------------------------
This path is subsumed by the proof of path {10}->{11,15}->{16}->{9}->{8}.
* Path {10}->{11,15}->{16}->{9}->{8}: NA
--------------------------------------
The usable rules for this path are:
{ check(0()) -> zero()
, check(s(0())) -> odd()
, check(s(s(0()))) -> even()
, check(s(s(s(x)))) -> check(s(x))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(x, y))
, p(s(x)) -> x
, p(0()) -> 0()
, half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
, if(zero(), x, y, z, u) -> z
, if(odd(), x, y, z, u) -> timesIter(p(x), y, u)
, if(even(), x, y, z, u) ->
plus(timesIter(half(x), y, half(z)),
timesIter(half(x), y, half(s(z))))}
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 {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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [1] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [3] x1 + [0]
c_11(x1) = [1] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {p^#(s(x)) -> c_11(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [7]
p^#(x1) = [1] x1 + [7]
c_11(x1) = [1] x1 + [1]
* Path {13}: 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(check) = {}, Uargs(s) = {}, Uargs(half) = {},
Uargs(plus) = {}, Uargs(times) = {}, Uargs(timesIter) = {},
Uargs(if) = {}, Uargs(p) = {}, Uargs(check^#) = {},
Uargs(c_3) = {}, Uargs(half^#) = {}, Uargs(c_6) = {},
Uargs(plus^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(times^#) = {}, Uargs(c_9) = {}, Uargs(timesIter^#) = {},
Uargs(c_10) = {}, Uargs(if^#) = {}, Uargs(p^#) = {},
Uargs(c_11) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
Uargs(c_15) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
check(x1) = [0] x1 + [0]
0() = [0]
zero() = [0]
s(x1) = [0] x1 + [0]
odd() = [0]
even() = [0]
half(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
times(x1, x2) = [0] x1 + [0] x2 + [0]
timesIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
if(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p(x1) = [0] x1 + [0]
check^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
times^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9(x1) = [0] x1 + [0]
timesIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4, x5) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0] x5 + [0]
p^#(x1) = [0] x1 + [0]
c_11(x1) = [0] x1 + [0]
c_12() = [0]
c_13(x1) = [0] x1 + [0]
c_14(x1) = [0] x1 + [0]
c_15(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: {p^#(0()) -> c_12()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_12() = [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.