Tool CaT
stdout:
MAYBE
Problem:
null(nil()) -> true()
null(add(n,x)) -> false()
tail(add(n,x)) -> x
tail(nil()) -> nil()
head(add(n,x)) -> n
app(nil(),y) -> y
app(add(n,x),y) -> add(n,app(x,y))
reverse(nil()) -> nil()
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
shuffle(x) -> shuff(x,nil())
shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil())))
if(true(),x,y,z) -> y
if(false(),x,y,z) -> shuff(reverse(tail(x)),z)
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ null(nil()) -> true()
, null(add(n, x)) -> false()
, tail(add(n, x)) -> x
, tail(nil()) -> nil()
, head(add(n, x)) -> n
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, shuffle(x) -> shuff(x, nil())
, shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil())))
, if(true(), x, y, z) -> y
, if(false(), x, y, z) -> shuff(reverse(tail(x)), z)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: null^#(nil()) -> c_0()
, 2: null^#(add(n, x)) -> c_1()
, 3: tail^#(add(n, x)) -> c_2()
, 4: tail^#(nil()) -> c_3()
, 5: head^#(add(n, x)) -> c_4()
, 6: app^#(nil(), y) -> c_5()
, 7: app^#(add(n, x), y) -> c_6(app^#(x, y))
, 8: reverse^#(nil()) -> c_7()
, 9: reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, 10: shuffle^#(x) -> c_9(shuff^#(x, nil()))
, 11: shuff^#(x, y) ->
c_10(if^#(null(x), x, y, app(y, add(head(x), nil()))))
, 12: if^#(true(), x, y, z) -> c_11()
, 13: if^#(false(), x, y, z) -> c_12(shuff^#(reverse(tail(x)), z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{8} [ 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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
app^#(x1, 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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {null^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
app^#(x1, 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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {null^#(add(n, x)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(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]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
app^#(x1, 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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(add(n, x)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(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_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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
app^#(x1, 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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
app^#(x1, 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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {head^#(add(n, x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(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_4() = [0]
[1]
[1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
app^#(x1, 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_5() = [0]
[0]
[0]
c_6(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11() = [0]
[0]
[0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(reverse^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
reverse^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_7() = [0]
[1]
[1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, app^#(nil(), y) -> c_5()
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ null(nil()) -> true()
, null(add(n, x)) -> false()
, tail(add(n, x)) -> x
, tail(nil()) -> nil()
, head(add(n, x)) -> n
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: null^#(nil()) -> c_0()
, 2: null^#(add(n, x)) -> c_1()
, 3: tail^#(add(n, x)) -> c_2()
, 4: tail^#(nil()) -> c_3()
, 5: head^#(add(n, x)) -> c_4()
, 6: app^#(nil(), y) -> c_5()
, 7: app^#(add(n, x), y) -> c_6(app^#(x, y))
, 8: reverse^#(nil()) -> c_7()
, 9: reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, 10: shuffle^#(x) -> c_9(shuff^#(x, nil()))
, 11: shuff^#(x, y) ->
c_10(if^#(null(x), x, y, app(y, add(head(x), nil()))))
, 12: if^#(true(), x, y, z) -> c_11()
, 13: if^#(false(), x, y, z) -> c_12(shuff^#(reverse(tail(x)), z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{8} [ 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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {null^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {null^#(add(n, x)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(add(n, x)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
tail^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {head^#(add(n, x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
head^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_4() = [0]
[1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11() = [0]
[0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(reverse^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
reverse^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_7() = [0]
[1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, app^#(nil(), y) -> c_5()
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ null(nil()) -> true()
, null(add(n, x)) -> false()
, tail(add(n, x)) -> x
, tail(nil()) -> nil()
, head(add(n, x)) -> n
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: null^#(nil()) -> c_0()
, 2: null^#(add(n, x)) -> c_1()
, 3: tail^#(add(n, x)) -> c_2()
, 4: tail^#(nil()) -> c_3()
, 5: head^#(add(n, x)) -> c_4()
, 6: app^#(nil(), y) -> c_5()
, 7: app^#(add(n, x), y) -> c_6(app^#(x, y))
, 8: reverse^#(nil()) -> c_7()
, 9: reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, 10: shuffle^#(x) -> c_9(shuff^#(x, nil()))
, 11: shuff^#(x, y) ->
c_10(if^#(null(x), x, y, app(y, add(head(x), nil()))))
, 12: if^#(true(), x, y, z) -> c_11()
, 13: if^#(false(), x, y, z) -> c_12(shuff^#(reverse(tail(x)), z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{8} [ 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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {null^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {null^#(add(n, x)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0] x1 + [0] x2 + [7]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {tail^#(add(n, x)) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(tail^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0] x1 + [0] x2 + [7]
tail^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {head^#(add(n, x)) -> c_4()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(head^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_4() = [1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(head^#) = {},
Uargs(app^#) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4() = [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11() = [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(reverse^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
reverse^#(x1) = [1] x1 + [7]
c_7() = [1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, app^#(nil(), y) -> c_5()
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ null(nil()) -> true()
, null(add(n, x)) -> false()
, tail(add(n, x)) -> x
, tail(nil()) -> nil()
, head(add(n, x)) -> n
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ null(nil()) -> true()
, null(add(n, x)) -> false()
, tail(add(n, x)) -> x
, tail(nil()) -> nil()
, head(add(n, x)) -> n
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, shuffle(x) -> shuff(x, nil())
, shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil())))
, if(true(), x, y, z) -> y
, if(false(), x, y, z) -> shuff(reverse(tail(x)), z)}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: null^#(nil()) -> c_0()
, 2: null^#(add(n, x)) -> c_1()
, 3: tail^#(add(n, x)) -> c_2(x)
, 4: tail^#(nil()) -> c_3()
, 5: head^#(add(n, x)) -> c_4(n)
, 6: app^#(nil(), y) -> c_5(y)
, 7: app^#(add(n, x), y) -> c_6(n, app^#(x, y))
, 8: reverse^#(nil()) -> c_7()
, 9: reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, 10: shuffle^#(x) -> c_9(shuff^#(x, nil()))
, 11: shuff^#(x, y) ->
c_10(if^#(null(x), x, y, app(y, add(head(x), nil()))))
, 12: if^#(true(), x, y, z) -> c_11(y)
, 13: if^#(false(), x, y, z) -> c_12(shuff^#(reverse(tail(x)), z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{8} [ 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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(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]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app^#(x1, 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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {null^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(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]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app^#(x1, 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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {null^#(add(n, x)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(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]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(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]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app^#(x1, 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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(add(n, x)) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(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_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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(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]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app^#(x1, 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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(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]
c_3() = [0]
[0]
[0]
head^#(x1) = [1 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app^#(x1, 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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(add(n, x)) -> c_4(n)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(head^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[0]
true() = [0]
[0]
[0]
add(x1, 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]
tail(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
head(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuff(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
if(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
null^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1() = [0]
[0]
[0]
tail^#(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]
c_3() = [0]
[0]
[0]
head^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
app^#(x1, 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_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
reverse^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_7() = [0]
[0]
[0]
c_8(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
shuffle^#(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]
shuff^#(x1, 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_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
if^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_12(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(reverse^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
[2]
reverse^#(x1) = [0 2 0] x1 + [7]
[2 2 0] [3]
[2 2 2] [3]
c_7() = [0]
[1]
[1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, app^#(nil(), y) -> c_5(y)
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ null(nil()) -> true()
, null(add(n, x)) -> false()
, tail(add(n, x)) -> x
, tail(nil()) -> nil()
, head(add(n, x)) -> n
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: null^#(nil()) -> c_0()
, 2: null^#(add(n, x)) -> c_1()
, 3: tail^#(add(n, x)) -> c_2(x)
, 4: tail^#(nil()) -> c_3()
, 5: head^#(add(n, x)) -> c_4(n)
, 6: app^#(nil(), y) -> c_5(y)
, 7: app^#(add(n, x), y) -> c_6(n, app^#(x, y))
, 8: reverse^#(nil()) -> c_7()
, 9: reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, 10: shuffle^#(x) -> c_9(shuff^#(x, nil()))
, 11: shuff^#(x, y) ->
c_10(if^#(null(x), x, y, app(y, add(head(x), nil()))))
, 12: if^#(true(), x, y, z) -> c_11(y)
, 13: if^#(false(), x, y, z) -> c_12(shuff^#(reverse(tail(x)), z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{8} [ 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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {null^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {null^#(add(n, x)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_2(x1) = [1 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(add(n, x)) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
tail^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
head^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(add(n, x)) -> c_4(n)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(head^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [1 2] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
head^#(x1) = [2 2] x1 + [7]
[2 0] [7]
c_4(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
true() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
null^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
tail^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3() = [0]
[0]
head^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
shuff^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
[0 0] [0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(reverse^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
reverse^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_7() = [0]
[1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, app^#(nil(), y) -> c_5(y)
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ null(nil()) -> true()
, null(add(n, x)) -> false()
, tail(add(n, x)) -> x
, tail(nil()) -> nil()
, head(add(n, x)) -> n
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: null^#(nil()) -> c_0()
, 2: null^#(add(n, x)) -> c_1()
, 3: tail^#(add(n, x)) -> c_2(x)
, 4: tail^#(nil()) -> c_3()
, 5: head^#(add(n, x)) -> c_4(n)
, 6: app^#(nil(), y) -> c_5(y)
, 7: app^#(add(n, x), y) -> c_6(n, app^#(x, y))
, 8: reverse^#(nil()) -> c_7()
, 9: reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, 10: shuffle^#(x) -> c_9(shuff^#(x, nil()))
, 11: shuff^#(x, y) ->
c_10(if^#(null(x), x, y, app(y, add(head(x), nil()))))
, 12: if^#(true(), x, y, z) -> c_11(y)
, 13: if^#(false(), x, y, z) -> c_12(shuff^#(reverse(tail(x)), z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{10} [ inherited ]
|
`->{11,13} [ inherited ]
|
`->{12} [ NA ]
->{9} [ inherited ]
|
|->{6} [ MAYBE ]
|
`->{7} [ inherited ]
|
`->{6} [ NA ]
->{8} [ 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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {null^#(nil()) -> c_0()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {null^#(add(n, x)) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(null^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0] x1 + [0] x2 + [7]
null^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [1] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [3] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {tail^#(add(n, x)) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [0] x1 + [1] x2 + [7]
tail^#(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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {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(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [1] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
head^#(x1) = [3] x1 + [0]
c_4(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {head^#(add(n, x)) -> c_4(n)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(add) = {}, Uargs(head^#) = {}, Uargs(c_4) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
add(x1, x2) = [1] x1 + [0] x2 + [7]
head^#(x1) = [1] x1 + [7]
c_4(x1) = [1] x1 + [1]
* Path {8}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(null) = {}, Uargs(add) = {}, Uargs(tail) = {},
Uargs(head) = {}, Uargs(app) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(shuff) = {}, Uargs(if) = {},
Uargs(null^#) = {}, Uargs(tail^#) = {}, Uargs(c_2) = {},
Uargs(head^#) = {}, Uargs(c_4) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(reverse^#) = {},
Uargs(c_8) = {}, Uargs(shuffle^#) = {}, Uargs(c_9) = {},
Uargs(shuff^#) = {}, Uargs(c_10) = {}, Uargs(if^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
null(x1) = [0] x1 + [0]
nil() = [0]
true() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
tail(x1) = [0] x1 + [0]
head(x1) = [0] x1 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
shuff(x1, x2) = [0] x1 + [0] x2 + [0]
if(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
null^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
tail^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
head^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
shuff^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
if^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_7()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(reverse^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
reverse^#(x1) = [1] x1 + [7]
c_7() = [1]
* Path {9}: inherited
-------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{6}: MAYBE
--------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ reverse^#(add(n, x)) -> c_8(app^#(reverse(x), add(n, nil())))
, app^#(nil(), y) -> c_5(y)
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {9}->{7}: inherited
------------------------
This path is subsumed by the proof of path {9}->{7}->{6}.
* Path {9}->{7}->{6}: NA
----------------------
The usable rules for this path are:
{ reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {10}: inherited
--------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}: inherited
-----------------------------
This path is subsumed by the proof of path {10}->{11,13}->{12}.
* Path {10}->{11,13}->{12}: NA
----------------------------
The usable rules for this path are:
{ null(nil()) -> true()
, null(add(n, x)) -> false()
, tail(add(n, x)) -> x
, tail(nil()) -> nil()
, head(add(n, x)) -> n
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, reverse(nil()) -> nil()
, reverse(add(n, x)) -> app(reverse(x), add(n, nil()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.