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