Tool CaT
stdout:
MAYBE
Problem:
empty(nil()) -> true()
empty(cons(x,l)) -> false()
head(cons(x,l)) -> x
tail(nil()) -> nil()
tail(cons(x,l)) -> l
rev(nil()) -> nil()
rev(cons(x,l)) -> cons(rev1(x,l),rev2(x,l))
last(x,l) -> if(empty(l),x,l)
if(true(),x,l) -> x
if(false(),x,l) -> last(head(l),tail(l))
rev2(x,nil()) -> nil()
rev2(x,cons(y,l)) -> rev(cons(x,rev2(y,l)))
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:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l
, rev(nil()) -> nil()
, rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
, last(x, l) -> if(empty(l), x, l)
, if(true(), x, l) -> x
, if(false(), x, l) -> last(head(l), tail(l))
, rev2(x, nil()) -> nil()
, rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))}
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: empty^#(nil()) -> c_0()
, 2: empty^#(cons(x, l)) -> c_1()
, 3: head^#(cons(x, l)) -> c_2()
, 4: tail^#(nil()) -> c_3()
, 5: tail^#(cons(x, l)) -> c_4()
, 6: rev^#(nil()) -> c_5()
, 7: rev^#(cons(x, l)) -> c_6(rev2^#(x, l))
, 8: last^#(x, l) -> c_7(if^#(empty(l), x, l))
, 9: if^#(true(), x, l) -> c_8()
, 10: if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, 11: rev2^#(x, nil()) -> c_10()
, 12: rev2^#(x, cons(y, l)) -> c_11(rev^#(cons(x, rev2(y, l))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,10} [ MAYBE ]
|
`->{9} [ NA ]
->{7,12} [ inherited ]
|
`->{11} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
empty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
empty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_1() = [0]
[1]
[1]
* 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {head^#(cons(x, l)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
head^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_2() = [0]
[1]
[1]
* Path {4}: 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {tail^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
tail^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {tail^#(cons(x, l)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
tail^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_4() = [0]
[1]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {rev^#(nil()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(rev^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
rev^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_5() = [0]
[1]
[1]
* Path {7,12}: inherited
----------------------
This path is subsumed by the proof of path {7,12}->{11}.
* Path {7,12}->{11}: NA
---------------------
The usable rules for this path are:
{ rev2(x, nil()) -> nil()
, rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
, rev(nil()) -> nil()
, rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,10}: MAYBE
------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
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:
{ last^#(x, l) -> c_7(if^#(empty(l), x, l))
, if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
Proof Output:
The input cannot be shown compatible
* Path {8,10}->{9}: NA
--------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {1, 2}, Uargs(c_7) = {1}, Uargs(if^#) = {1},
Uargs(c_9) = {1}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [3]
nil() = [2]
[0]
[0]
true() = [1]
[1]
[1]
cons(x1, x2) = [1 1 0] x1 + [1 3 1] x2 + [2]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 0] [0]
false() = [1]
[1]
[1]
head(x1) = [3 3 3] x1 + [3]
[3 3 3] [3]
[0 0 1] [3]
tail(x1) = [2 0 0] x1 + [0]
[0 2 0] [0]
[1 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4() = [0]
[0]
[0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
if^#(x1, x2, x3) = [3 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8() = [0]
[0]
[0]
c_9(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_10() = [0]
[0]
[0]
c_11(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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: empty^#(nil()) -> c_0()
, 2: empty^#(cons(x, l)) -> c_1()
, 3: head^#(cons(x, l)) -> c_2()
, 4: tail^#(nil()) -> c_3()
, 5: tail^#(cons(x, l)) -> c_4()
, 6: rev^#(nil()) -> c_5()
, 7: rev^#(cons(x, l)) -> c_6(rev2^#(x, l))
, 8: last^#(x, l) -> c_7(if^#(empty(l), x, l))
, 9: if^#(true(), x, l) -> c_8()
, 10: if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, 11: rev2^#(x, nil()) -> c_10()
, 12: rev2^#(x, cons(y, l)) -> c_11(rev^#(cons(x, rev2(y, l))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,10} [ MAYBE ]
|
`->{9} [ NA ]
->{7,12} [ inherited ]
|
`->{11} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
empty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
empty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_1() = [0]
[1]
* 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {head^#(cons(x, l)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
head^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_2() = [0]
[1]
* Path {4}: 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {tail^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
tail^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {tail^#(cons(x, l)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
tail^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_4() = [0]
[1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {rev^#(nil()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(rev^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
rev^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_5() = [0]
[1]
* Path {7,12}: inherited
----------------------
This path is subsumed by the proof of path {7,12}->{11}.
* Path {7,12}->{11}: NA
---------------------
The usable rules for this path are:
{ rev2(x, nil()) -> nil()
, rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
, rev(nil()) -> nil()
, rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,10}: MAYBE
------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
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:
{ last^#(x, l) -> c_7(if^#(empty(l), x, l))
, if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
Proof Output:
The input cannot be shown compatible
* Path {8,10}->{9}: NA
--------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {1, 2}, Uargs(c_7) = {1}, Uargs(if^#) = {1},
Uargs(c_9) = {1}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [2]
[3 3] [3]
nil() = [2]
[0]
true() = [1]
[1]
cons(x1, x2) = [1 1] x1 + [1 3] x2 + [1]
[0 1] [0 1] [0]
false() = [1]
[0]
head(x1) = [3 3] x1 + [3]
[3 3] [3]
tail(x1) = [2 0] x1 + [0]
[0 2] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4() = [0]
[0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
c_10() = [0]
[0]
c_11(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.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: empty^#(nil()) -> c_0()
, 2: empty^#(cons(x, l)) -> c_1()
, 3: head^#(cons(x, l)) -> c_2()
, 4: tail^#(nil()) -> c_3()
, 5: tail^#(cons(x, l)) -> c_4()
, 6: rev^#(nil()) -> c_5()
, 7: rev^#(cons(x, l)) -> c_6(rev2^#(x, l))
, 8: last^#(x, l) -> c_7(if^#(empty(l), x, l))
, 9: if^#(true(), x, l) -> c_8()
, 10: if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, 11: rev2^#(x, nil()) -> c_10()
, 12: rev2^#(x, cons(y, l)) -> c_11(rev^#(cons(x, rev2(y, l))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,10} [ MAYBE ]
|
`->{9} [ NA ]
->{7,12} [ inherited ]
|
`->{11} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
empty^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
empty^#(x1) = [1] x1 + [7]
c_1() = [1]
* 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {head^#(cons(x, l)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_2() = [1]
* Path {4}: 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {tail^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
tail^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {tail^#(cons(x, l)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_4() = [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {rev^#(nil()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(rev^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
rev^#(x1) = [1] x1 + [7]
c_5() = [1]
* Path {7,12}: inherited
----------------------
This path is subsumed by the proof of path {7,12}->{11}.
* Path {7,12}->{11}: NA
---------------------
The usable rules for this path are:
{ rev2(x, nil()) -> nil()
, rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
, rev(nil()) -> nil()
, rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,10}: MAYBE
------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
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:
{ last^#(x, l) -> c_7(if^#(empty(l), x, l))
, if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
Proof Output:
The input cannot be shown compatible
* Path {8,10}->{9}: NA
--------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
Uargs(rev^#) = {}, Uargs(c_6) = {}, Uargs(rev2^#) = {},
Uargs(last^#) = {1, 2}, Uargs(c_7) = {1}, Uargs(if^#) = {1},
Uargs(c_9) = {1}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [3] x1 + [3]
nil() = [1]
true() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [1]
false() = [1]
head(x1) = [3] x1 + [3]
tail(x1) = [3] x1 + [3]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2() = [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4() = [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_7(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_8() = [0]
c_9(x1) = [1] x1 + [0]
c_10() = [0]
c_11(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.
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:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l
, rev(nil()) -> nil()
, rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
, last(x, l) -> if(empty(l), x, l)
, if(true(), x, l) -> x
, if(false(), x, l) -> last(head(l), tail(l))
, rev2(x, nil()) -> nil()
, rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))}
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: empty^#(nil()) -> c_0()
, 2: empty^#(cons(x, l)) -> c_1()
, 3: head^#(cons(x, l)) -> c_2(x)
, 4: tail^#(nil()) -> c_3()
, 5: tail^#(cons(x, l)) -> c_4(l)
, 6: rev^#(nil()) -> c_5()
, 7: rev^#(cons(x, l)) -> c_6(x, l, rev2^#(x, l))
, 8: last^#(x, l) -> c_7(if^#(empty(l), x, l))
, 9: if^#(true(), x, l) -> c_8(x)
, 10: if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, 11: rev2^#(x, nil()) -> c_10()
, 12: rev2^#(x, cons(y, l)) -> c_11(rev^#(cons(x, rev2(y, l))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,10} [ MAYBE ]
|
`->{9} [ NA ]
->{7,12} [ inherited ]
|
`->{11} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^3)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^3)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
empty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_0() = [0]
[1]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
[0 0 0] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
empty^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_1() = [0]
[1]
[1]
* Path {3}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
[0 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(cons(x, l)) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [0 0 0] x2 + [2]
[0 0 2] [0 0 0] [2]
[0 0 0] [0 0 0] [2]
head^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {4}: 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {tail^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
tail^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_3() = [0]
[1]
[1]
* Path {5}: YES(?,O(n^3))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
[0 1 1] [0 0 0] [0]
[0 0 1] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(x, l)) -> c_4(l)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
[0 0 0] [0 0 2] [2]
[0 0 0] [0 0 0] [2]
tail^#(x1) = [2 2 2] x1 + [3]
[2 2 2] [3]
[2 2 2] [3]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
false() = [0]
[0]
[0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, 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_7(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_10() = [0]
[0]
[0]
c_11(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: {rev^#(nil()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(rev^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
rev^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_5() = [0]
[1]
[1]
* Path {7,12}: inherited
----------------------
This path is subsumed by the proof of path {7,12}->{11}.
* Path {7,12}->{11}: NA
---------------------
The usable rules for this path are:
{ rev2(x, nil()) -> nil()
, rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
, rev(nil()) -> nil()
, rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,10}: MAYBE
------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
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:
{ last^#(x, l) -> c_7(if^#(empty(l), x, l))
, if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
Proof Output:
The input cannot be shown compatible
* Path {8,10}->{9}: NA
--------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {1}, Uargs(cons) = {}, Uargs(head) = {1},
Uargs(tail) = {1}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {1, 2},
Uargs(c_7) = {1}, Uargs(if^#) = {1, 2, 3}, Uargs(c_8) = {1},
Uargs(c_9) = {1}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [2 0 0] x1 + [2]
[0 0 0] [3]
[0 0 0] [3]
nil() = [0]
[2]
[0]
true() = [1]
[1]
[1]
cons(x1, x2) = [1 1 0] x1 + [1 3 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 0 1] [0 0 1] [0]
false() = [1]
[1]
[1]
head(x1) = [2 0 0] x1 + [3]
[3 3 3] [3]
[0 0 1] [3]
tail(x1) = [2 0 0] x1 + [2]
[0 2 0] [0]
[0 0 1] [0]
rev(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
rev2(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last(x1, 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) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
empty^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
rev^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5() = [0]
[0]
[0]
c_6(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
rev2^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
last^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_7(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
if^#(x1, x2, x3) = [3 0 0] x1 + [3 3 3] x2 + [3 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_8(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_9(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_10() = [0]
[0]
[0]
c_11(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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: empty^#(nil()) -> c_0()
, 2: empty^#(cons(x, l)) -> c_1()
, 3: head^#(cons(x, l)) -> c_2(x)
, 4: tail^#(nil()) -> c_3()
, 5: tail^#(cons(x, l)) -> c_4(l)
, 6: rev^#(nil()) -> c_5()
, 7: rev^#(cons(x, l)) -> c_6(x, l, rev2^#(x, l))
, 8: last^#(x, l) -> c_7(if^#(empty(l), x, l))
, 9: if^#(true(), x, l) -> c_8(x)
, 10: if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, 11: rev2^#(x, nil()) -> c_10()
, 12: rev2^#(x, cons(y, l)) -> c_11(rev^#(cons(x, rev2(y, l))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,10} [ MAYBE ]
|
`->{9} [ NA ]
->{7,12} [ inherited ]
|
`->{11} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^2)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
empty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_0() = [0]
[1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
empty^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_1() = [0]
[1]
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(cons(x, l)) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
head^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_2(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {4}: 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {tail^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
tail^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_3() = [0]
[1]
* Path {5}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 0] [0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(x, l)) -> c_4(l)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
tail^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_4(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
c_10() = [0]
[0]
c_11(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: {rev^#(nil()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(rev^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
rev^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_5() = [0]
[1]
* Path {7,12}: inherited
----------------------
This path is subsumed by the proof of path {7,12}->{11}.
* Path {7,12}->{11}: NA
---------------------
The usable rules for this path are:
{ rev2(x, nil()) -> nil()
, rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
, rev(nil()) -> nil()
, rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,10}: MAYBE
------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
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:
{ last^#(x, l) -> c_7(if^#(empty(l), x, l))
, if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
Proof Output:
The input cannot be shown compatible
* Path {8,10}->{9}: NA
--------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {1}, Uargs(cons) = {}, Uargs(head) = {1},
Uargs(tail) = {1}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {1, 2},
Uargs(c_7) = {1}, Uargs(if^#) = {1, 2, 3}, Uargs(c_8) = {1},
Uargs(c_9) = {1}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [2 0] x1 + [3]
[3 3] [3]
nil() = [2]
[0]
true() = [0]
[1]
cons(x1, x2) = [1 1] x1 + [1 3] x2 + [2]
[0 1] [0 1] [0]
false() = [0]
[1]
head(x1) = [3 3] x1 + [3]
[3 3] [3]
tail(x1) = [2 0] x1 + [0]
[0 2] [0]
rev(x1) = [0 0] x1 + [0]
[0 0] [0]
rev1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
rev2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
empty^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
rev^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
c_6(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
rev2^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
last^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
if^#(x1, x2, x3) = [3 0] x1 + [3 3] x2 + [3 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
c_9(x1) = [1 0] x1 + [0]
[0 1] [0]
c_10() = [0]
[0]
c_11(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.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: empty^#(nil()) -> c_0()
, 2: empty^#(cons(x, l)) -> c_1()
, 3: head^#(cons(x, l)) -> c_2(x)
, 4: tail^#(nil()) -> c_3()
, 5: tail^#(cons(x, l)) -> c_4(l)
, 6: rev^#(nil()) -> c_5()
, 7: rev^#(cons(x, l)) -> c_6(x, l, rev2^#(x, l))
, 8: last^#(x, l) -> c_7(if^#(empty(l), x, l))
, 9: if^#(true(), x, l) -> c_8(x)
, 10: if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, 11: rev2^#(x, nil()) -> c_10()
, 12: rev2^#(x, cons(y, l)) -> c_11(rev^#(cons(x, rev2(y, l))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{8,10} [ MAYBE ]
|
`->{9} [ NA ]
->{7,12} [ inherited ]
|
`->{11} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(1)) ]
->{3} [ YES(?,O(n^1)) ]
->{2} [ YES(?,O(1)) ]
->{1} [ YES(?,O(1)) ]
Sub-problems:
-------------
* Path {1}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {empty^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
empty^#(x1) = [1] x1 + [7]
c_0() = [1]
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {empty^#(cons(x, l)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(empty^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [0] x2 + [7]
empty^#(x1) = [1] x1 + [7]
c_1() = [1]
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [3] x1 + [0]
c_2(x1) = [1] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(cons(x, l)) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_2(x1) = [1] x1 + [1]
* Path {4}: 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(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {tail^#(nil()) -> c_3()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
tail^#(x1) = [1] x1 + [7]
c_3() = [1]
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [3] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(cons(x, l)) -> c_4(l)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [7]
tail^#(x1) = [1] x1 + [7]
c_4(x1) = [1] x1 + [1]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {}, Uargs(cons) = {}, Uargs(head) = {},
Uargs(tail) = {}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {},
Uargs(c_7) = {}, Uargs(if^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
head(x1) = [0] x1 + [0]
tail(x1) = [0] x1 + [0]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
c_10() = [0]
c_11(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: {rev^#(nil()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(rev^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
rev^#(x1) = [1] x1 + [7]
c_5() = [1]
* Path {7,12}: inherited
----------------------
This path is subsumed by the proof of path {7,12}->{11}.
* Path {7,12}->{11}: NA
---------------------
The usable rules for this path are:
{ rev2(x, nil()) -> nil()
, rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
, rev(nil()) -> nil()
, rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {8,10}: MAYBE
------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
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:
{ last^#(x, l) -> c_7(if^#(empty(l), x, l))
, if^#(false(), x, l) -> c_9(last^#(head(l), tail(l)))
, empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
Proof Output:
The input cannot be shown compatible
* Path {8,10}->{9}: NA
--------------------
The usable rules for this path are:
{ empty(nil()) -> true()
, empty(cons(x, l)) -> false()
, head(cons(x, l)) -> x
, tail(nil()) -> nil()
, tail(cons(x, l)) -> l}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(empty) = {1}, Uargs(cons) = {}, Uargs(head) = {1},
Uargs(tail) = {1}, Uargs(rev) = {}, Uargs(rev1) = {},
Uargs(rev2) = {}, Uargs(last) = {}, Uargs(if) = {},
Uargs(empty^#) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
Uargs(tail^#) = {}, Uargs(c_4) = {}, Uargs(rev^#) = {},
Uargs(c_6) = {}, Uargs(rev2^#) = {}, Uargs(last^#) = {1, 2},
Uargs(c_7) = {1}, Uargs(if^#) = {1, 2, 3}, Uargs(c_8) = {1},
Uargs(c_9) = {1}, Uargs(c_11) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
empty(x1) = [1] x1 + [3]
nil() = [3]
true() = [1]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
false() = [1]
head(x1) = [1] x1 + [3]
tail(x1) = [1] x1 + [3]
rev(x1) = [0] x1 + [0]
rev1(x1, x2) = [0] x1 + [0] x2 + [0]
rev2(x1, x2) = [0] x1 + [0] x2 + [0]
last(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
empty^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
head^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
tail^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
rev^#(x1) = [0] x1 + [0]
c_5() = [0]
c_6(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
rev2^#(x1, x2) = [0] x1 + [0] x2 + [0]
last^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_7(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [3] x1 + [3] x2 + [3] x3 + [0]
c_8(x1) = [1] x1 + [0]
c_9(x1) = [1] x1 + [0]
c_10() = [0]
c_11(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.
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.