Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.53 |
|---|
stdout:
MAYBE
Problem:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
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(nil()) -> nil()
shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
concat(leaf(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
less_leaves(x,leaf()) -> false()
less_leaves(leaf(),cons(w,z)) -> true()
less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.53 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.53 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, 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(nil()) -> nil()
, shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))
, less_leaves(x, leaf()) -> false()
, less_leaves(leaf(), cons(w, z)) -> true()
, less_leaves(cons(u, v), cons(w, z)) ->
less_leaves(concat(u, v), concat(w, 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: minus^#(x, 0()) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: app^#(nil(), y) -> c_4()
, 6: app^#(add(n, x), y) -> c_5(app^#(x, y))
, 7: reverse^#(nil()) -> c_6()
, 8: reverse^#(add(n, x)) -> c_7(app^#(reverse(x), add(n, nil())))
, 9: shuffle^#(nil()) -> c_8()
, 10: shuffle^#(add(n, x)) -> c_9(shuffle^#(reverse(x)))
, 11: concat^#(leaf(), y) -> c_10()
, 12: concat^#(cons(u, v), y) -> c_11(concat^#(v, y))
, 13: less_leaves^#(x, leaf()) -> c_12()
, 14: less_leaves^#(leaf(), cons(w, z)) -> c_13()
, 15: less_leaves^#(cons(u, v), cons(w, z)) ->
c_14(less_leaves^#(concat(u, v), concat(w, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ NA ]
|
|->{13} [ NA ]
|
`->{14} [ NA ]
->{12} [ YES(?,O(n^3)) ]
|
`->{11} [ YES(?,O(n^3)) ]
->{10} [ inherited ]
|
`->{9} [ MAYBE ]
->{8} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
->{7} [ YES(?,O(1)) ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^3)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
minus^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_0()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 3] [2]
[0 0 1] [2]
minus^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
[0 0 2] [2 2 0] [0]
[0 0 0] [0 2 2] [0]
c_0() = [1]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [7]
* Path {4}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [3]
[0 2 0] [0 0 0] [1]
[0 0 1] [0 0 0] [1]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [1]
[0 1 0] [0]
[0 0 1] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2 0 3] x1 + [0 0 1] x2 + [3]
[0 2 2] [0 0 0] [1]
[0 0 1] [0 0 0] [3]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [2]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_6()}
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_6() = [0]
[1]
[1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{5}: 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 {8}->{6}: inherited
------------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{6}->{5}: 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}->{9}.
* Path {10}->{9}: 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:
{ shuffle^#(add(n, x)) -> c_9(shuffle^#(reverse(x)))
, shuffle^#(nil()) -> c_8()
, 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 {12}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {1}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_10() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {concat^#(cons(u, v), y) -> c_11(concat^#(v, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
[0 0 0] [0 1 2] [2]
[0 0 0] [0 0 1] [2]
concat^#(x1, x2) = [0 1 2] x1 + [0 0 0] x2 + [2]
[2 2 2] [4 4 4] [0]
[2 0 2] [4 4 4] [0]
c_11(x1) = [1 0 0] x1 + [3]
[0 0 0] [6]
[2 0 0] [2]
* Path {12}->{11}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {1}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {concat^#(leaf(), y) -> c_10()}
Weak Rules: {concat^#(cons(u, v), y) -> c_11(concat^#(v, y))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
leaf() = [2]
[2]
[2]
cons(x1, x2) = [0 0 0] x1 + [1 4 0] x2 + [0]
[0 0 0] [0 1 3] [2]
[0 0 0] [0 0 1] [2]
concat^#(x1, x2) = [0 3 2] x1 + [0 0 0] x2 + [0]
[2 2 2] [4 4 0] [0]
[2 2 2] [4 4 0] [0]
c_10() = [1]
[0]
[0]
c_11(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {15}: NA
-------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 3] x1 + [1 0 0] x2 + [2]
[0 0 1] [0 1 0] [0]
[0 0 2] [3 0 2] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 3] x1 + [1 0 0] x2 + [0]
[0 0 1] [0 1 0] [0]
[0 0 1] [0 0 1] [2]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
less_leaves^#(x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{13}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [3 0 0] x1 + [3 2 2] x2 + [2]
[2 0 0] [3 3 3] [0]
[2 0 0] [3 3 3] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
less_leaves^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{14}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [3 0 0] x1 + [3 2 2] x2 + [2]
[2 0 0] [3 3 3] [0]
[2 0 0] [3 3 3] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4() = [0]
[0]
[0]
c_5(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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
concat^#(x1, 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() = [0]
[0]
[0]
c_11(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
less_leaves^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: minus^#(x, 0()) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: app^#(nil(), y) -> c_4()
, 6: app^#(add(n, x), y) -> c_5(app^#(x, y))
, 7: reverse^#(nil()) -> c_6()
, 8: reverse^#(add(n, x)) -> c_7(app^#(reverse(x), add(n, nil())))
, 9: shuffle^#(nil()) -> c_8()
, 10: shuffle^#(add(n, x)) -> c_9(shuffle^#(reverse(x)))
, 11: concat^#(leaf(), y) -> c_10()
, 12: concat^#(cons(u, v), y) -> c_11(concat^#(v, y))
, 13: less_leaves^#(x, leaf()) -> c_12()
, 14: less_leaves^#(leaf(), cons(w, z)) -> c_13()
, 15: less_leaves^#(cons(u, v), cons(w, z)) ->
c_14(less_leaves^#(concat(u, v), concat(w, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ NA ]
|
|->{13} [ NA ]
|
`->{14} [ NA ]
->{12} [ YES(?,O(n^1)) ]
|
`->{11} [ YES(?,O(n^1)) ]
->{10} [ inherited ]
|
`->{9} [ MAYBE ]
->{8} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
->{7} [ YES(?,O(1)) ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ YES(?,O(n^2)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
minus^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_1(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_0()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
minus^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
[0 0] [4 1] [0]
c_0() = [1]
[0]
c_1(x1) = [1 0] x1 + [6]
[0 0] [7]
* Path {4}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 1] x1 + [0 0] x2 + [3]
[0 1] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 1] x1 + [0]
[0 1] [1]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
[0 2] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_6()}
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_6() = [0]
[1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{5}: 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 {8}->{6}: inherited
------------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{6}->{5}: 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}->{9}.
* Path {10}->{9}: 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:
{ shuffle^#(add(n, x)) -> c_9(shuffle^#(reverse(x)))
, shuffle^#(nil()) -> c_8()
, 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 {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {1}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {concat^#(cons(u, v), y) -> c_11(concat^#(v, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [2]
concat^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[4 5] [4 4] [0]
c_11(x1) = [1 0] x1 + [1]
[0 0] [7]
* Path {12}->{11}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {1}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {concat^#(leaf(), y) -> c_10()}
Weak Rules: {concat^#(cons(u, v), y) -> c_11(concat^#(v, y))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
leaf() = [2]
[0]
cons(x1, x2) = [0 0] x1 + [1 3] x2 + [2]
[0 0] [0 0] [2]
concat^#(x1, x2) = [3 0] x1 + [0 0] x2 + [4]
[2 2] [0 4] [0]
c_10() = [1]
[0]
c_11(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {15}: NA
-------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[0 1] [3 2] [0]
leaf() = [0]
[0]
cons(x1, x2) = [1 3] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
less_leaves^#(x1, x2) = [1 0] x1 + [2 0] x2 + [0]
[3 3] [3 3] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{13}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [2 0] x1 + [3 2] x2 + [1]
[0 0] [3 3] [0]
leaf() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
less_leaves^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{14}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [2 0] x1 + [3 2] x2 + [1]
[0 0] [3 3] [0]
leaf() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [0 0] x1 + [0]
[0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10() = [0]
[0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
less_leaves^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: minus^#(x, 0()) -> c_0()
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: app^#(nil(), y) -> c_4()
, 6: app^#(add(n, x), y) -> c_5(app^#(x, y))
, 7: reverse^#(nil()) -> c_6()
, 8: reverse^#(add(n, x)) -> c_7(app^#(reverse(x), add(n, nil())))
, 9: shuffle^#(nil()) -> c_8()
, 10: shuffle^#(add(n, x)) -> c_9(shuffle^#(reverse(x)))
, 11: concat^#(leaf(), y) -> c_10()
, 12: concat^#(cons(u, v), y) -> c_11(concat^#(v, y))
, 13: less_leaves^#(x, leaf()) -> c_12()
, 14: less_leaves^#(leaf(), cons(w, z)) -> c_13()
, 15: less_leaves^#(cons(u, v), cons(w, z)) ->
c_14(less_leaves^#(concat(u, v), concat(w, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ MAYBE ]
|
|->{13} [ NA ]
|
`->{14} [ NA ]
->{12} [ YES(?,O(n^1)) ]
|
`->{11} [ YES(?,O(n^1)) ]
->{10} [ inherited ]
|
`->{9} [ NA ]
->{8} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
->{7} [ YES(?,O(1)) ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_0()}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [4]
c_0() = [1]
c_1(x1) = [1] x1 + [7]
* Path {4}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_6()}
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_6() = [1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{5}: 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 {8}->{6}: inherited
------------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{6}->{5}: 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}->{9}.
* Path {10}->{9}: 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 {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {1}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [1] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {concat^#(cons(u, v), y) -> c_11(concat^#(v, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [0] x1 + [1] x2 + [4]
concat^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_11(x1) = [1] x1 + [7]
* Path {12}->{11}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {1}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [1] x1 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {concat^#(leaf(), y) -> c_10()}
Weak Rules: {concat^#(cons(u, v), y) -> c_11(concat^#(v, y))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
leaf() = [2]
cons(x1, x2) = [0] x1 + [1] x2 + [2]
concat^#(x1, x2) = [2] x1 + [7] x2 + [4]
c_10() = [1]
c_11(x1) = [1] x1 + [4]
* Path {15}: MAYBE
----------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, 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:
{ less_leaves^#(cons(u, v), cons(w, z)) ->
c_14(less_leaves^#(concat(u, v), concat(w, z)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
Proof Output:
The input cannot be shown compatible
* Path {15}->{13}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [2] x1 + [3] x2 + [1]
leaf() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
less_leaves^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{14}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(app^#) = {},
Uargs(c_5) = {}, Uargs(reverse^#) = {}, Uargs(c_7) = {},
Uargs(shuffle^#) = {}, Uargs(c_9) = {}, Uargs(concat^#) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [2] x1 + [3] x2 + [1]
leaf() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10() = [0]
c_11(x1) = [0] x1 + [0]
less_leaves^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.53 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.53 |
|---|
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, 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(nil()) -> nil()
, shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))
, less_leaves(x, leaf()) -> false()
, less_leaves(leaf(), cons(w, z)) -> true()
, less_leaves(cons(u, v), cons(w, z)) ->
less_leaves(concat(u, v), concat(w, 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: minus^#(x, 0()) -> c_0(x)
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: app^#(nil(), y) -> c_4(y)
, 6: app^#(add(n, x), y) -> c_5(n, app^#(x, y))
, 7: reverse^#(nil()) -> c_6()
, 8: reverse^#(add(n, x)) -> c_7(app^#(reverse(x), add(n, nil())))
, 9: shuffle^#(nil()) -> c_8()
, 10: shuffle^#(add(n, x)) -> c_9(n, shuffle^#(reverse(x)))
, 11: concat^#(leaf(), y) -> c_10(y)
, 12: concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))
, 13: less_leaves^#(x, leaf()) -> c_12()
, 14: less_leaves^#(leaf(), cons(w, z)) -> c_13()
, 15: less_leaves^#(cons(u, v), cons(w, z)) ->
c_14(less_leaves^#(concat(u, v), concat(w, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ NA ]
|
|->{13} [ NA ]
|
`->{14} [ NA ]
->{12} [ YES(?,O(n^3)) ]
|
`->{11} [ YES(?,O(n^1)) ]
->{10} [ inherited ]
|
`->{9} [ MAYBE ]
->{8} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
->{7} [ YES(?,O(1)) ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, 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]
c_11(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
minus^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, 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]
c_11(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_0(x)}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 1] x1 + [0]
[0 0 4] [2]
[0 0 1] [2]
minus^#(x1, x2) = [2 0 2] x1 + [2 0 2] x2 + [0]
[0 2 0] [0 3 2] [0]
[4 2 2] [5 2 0] [0]
c_0(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [6]
[0 0 0] [3]
[2 3 0] [7]
* Path {4}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [3]
[0 2 0] [0 0 0] [1]
[0 0 1] [0 0 0] [1]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [1]
[0 1 0] [0]
[0 0 1] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, 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]
c_11(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2 0 3] x1 + [0 0 1] x2 + [3]
[0 2 2] [0 0 0] [1]
[0 0 1] [0 0 0] [3]
0() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [2]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, 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]
c_11(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, 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]
c_11(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_6()}
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_6() = [0]
[1]
[1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{5}: 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 {8}->{6}: inherited
------------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{6}->{5}: 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}->{9}.
* Path {10}->{9}: 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:
{ shuffle^#(add(n, x)) -> c_9(n, shuffle^#(reverse(x)))
, shuffle^#(nil()) -> c_8()
, 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 {12}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
[0 1 3] [0 1 0] [0]
[0 0 1] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_10(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2 2] x1 + [1 6 0] x2 + [2]
[0 0 2] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
[2 2 0] [4 4 0] [0]
[0 0 0] [4 0 4] [0]
c_11(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [1]
[0 0 0] [0 0 0] [3]
[0 0 0] [0 0 0] [0]
* Path {12}->{11}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_10(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_11(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[0 0 0] [0 0 1] [0]
less_leaves^#(x1, 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_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {concat^#(leaf(), y) -> c_10(y)}
Weak Rules: {concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
leaf() = [2]
[2]
[2]
cons(x1, x2) = [1 0 0] x1 + [1 0 4] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [2]
concat^#(x1, x2) = [2 0 2] x1 + [0 0 0] x2 + [0]
[2 2 2] [4 0 0] [0]
[2 0 2] [4 0 0] [0]
c_10(x1) = [0 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_11(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [7]
[0 0 0] [0 0 0] [7]
[0 0 0] [0 0 0] [7]
* Path {15}: NA
-------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {2}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [1 0 1] x1 + [1 0 0] x2 + [0]
[0 0 0] [3 1 0] [0]
[0 0 1] [3 0 1] [2]
leaf() = [0]
[0]
[1]
cons(x1, x2) = [1 0 1] x1 + [1 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 1] [2]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, 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]
c_11(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves^#(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [0]
c_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{13}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {2}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [2 0 0] x1 + [3 2 3] x2 + [2]
[0 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, 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]
c_11(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{14}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {2}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 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]
nil() = [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]
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]
concat(x1, x2) = [2 0 0] x1 + [3 2 3] x2 + [2]
[0 0 0] [3 3 3] [0]
[0 0 0] [3 3 3] [0]
leaf() = [0]
[0]
[0]
cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [2]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves(x1, 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]
true() = [0]
[0]
[0]
minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
quot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
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_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1, 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_6() = [0]
[0]
[0]
c_7(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_8() = [0]
[0]
[0]
c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
concat^#(x1, 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]
c_11(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
less_leaves^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_12() = [0]
[0]
[0]
c_13() = [0]
[0]
[0]
c_14(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: minus^#(x, 0()) -> c_0(x)
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: app^#(nil(), y) -> c_4(y)
, 6: app^#(add(n, x), y) -> c_5(n, app^#(x, y))
, 7: reverse^#(nil()) -> c_6()
, 8: reverse^#(add(n, x)) -> c_7(app^#(reverse(x), add(n, nil())))
, 9: shuffle^#(nil()) -> c_8()
, 10: shuffle^#(add(n, x)) -> c_9(n, shuffle^#(reverse(x)))
, 11: concat^#(leaf(), y) -> c_10(y)
, 12: concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))
, 13: less_leaves^#(x, leaf()) -> c_12()
, 14: less_leaves^#(leaf(), cons(w, z)) -> c_13()
, 15: less_leaves^#(cons(u, v), cons(w, z)) ->
c_14(less_leaves^#(concat(u, v), concat(w, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ NA ]
|
|->{13} [ NA ]
|
`->{14} [ NA ]
->{12} [ YES(?,O(n^2)) ]
|
`->{11} [ YES(?,O(n^1)) ]
->{10} [ inherited ]
|
`->{9} [ MAYBE ]
->{8} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
->{7} [ YES(?,O(1)) ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ YES(?,O(n^2)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
minus^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_1(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 1] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_0(x)}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
minus^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
[4 1] [3 2] [0]
c_0(x1) = [0 0] x1 + [1]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[2 0] [0]
* Path {4}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 1] x1 + [0 0] x2 + [3]
[0 1] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 1] x1 + [0]
[0 1] [1]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
[0 2] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_6()}
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_6() = [0]
[1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{5}: 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 {8}->{6}: inherited
------------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{6}->{5}: 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}->{9}.
* Path {10}->{9}: 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:
{ shuffle^#(add(n, x)) -> c_9(n, shuffle^#(reverse(x)))
, shuffle^#(nil()) -> c_8()
, 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 {12}: YES(?,O(n^2))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [1 1] x1 + [1 2] x2 + [0]
[0 1] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1 2] x1 + [1 0] x2 + [2]
[0 0] [0 1] [0]
concat^#(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[2 1] [0 4] [0]
c_11(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [3]
* Path {12}->{11}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
leaf() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [1 1] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
less_leaves^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {concat^#(leaf(), y) -> c_10(y)}
Weak Rules: {concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
leaf() = [2]
[2]
cons(x1, x2) = [1 7] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
concat^#(x1, x2) = [2 2] x1 + [0 0] x2 + [0]
[2 2] [4 4] [0]
c_10(x1) = [0 0] x1 + [1]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [1 0] x2 + [6]
[0 0] [0 0] [7]
* Path {15}: NA
-------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {2}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 2] [0 1] [0]
leaf() = [1]
[0]
cons(x1, x2) = [1 3] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves^#(x1, x2) = [2 3] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{13}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {2}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [2 0] x1 + [3 3] x2 + [2]
[0 0] [3 3] [0]
leaf() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{14}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {2}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(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]
concat(x1, x2) = [2 0] x1 + [3 3] x2 + [2]
[0 0] [3 3] [0]
leaf() = [0]
[0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
less_leaves(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
false() = [0]
[0]
true() = [0]
[0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
reverse^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_6() = [0]
[0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
shuffle^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8() = [0]
[0]
c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
concat^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
less_leaves^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_12() = [0]
[0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
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: minus^#(x, 0()) -> c_0(x)
, 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
, 3: quot^#(0(), s(y)) -> c_2()
, 4: quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, 5: app^#(nil(), y) -> c_4(y)
, 6: app^#(add(n, x), y) -> c_5(n, app^#(x, y))
, 7: reverse^#(nil()) -> c_6()
, 8: reverse^#(add(n, x)) -> c_7(app^#(reverse(x), add(n, nil())))
, 9: shuffle^#(nil()) -> c_8()
, 10: shuffle^#(add(n, x)) -> c_9(n, shuffle^#(reverse(x)))
, 11: concat^#(leaf(), y) -> c_10(y)
, 12: concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))
, 13: less_leaves^#(x, leaf()) -> c_12()
, 14: less_leaves^#(leaf(), cons(w, z)) -> c_13()
, 15: less_leaves^#(cons(u, v), cons(w, z)) ->
c_14(less_leaves^#(concat(u, v), concat(w, z)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ NA ]
|
|->{13} [ NA ]
|
`->{14} [ NA ]
->{12} [ YES(?,O(n^1)) ]
|
`->{11} [ YES(?,O(n^1)) ]
->{10} [ inherited ]
|
`->{9} [ MAYBE ]
->{8} [ inherited ]
|
|->{5} [ NA ]
|
`->{6} [ inherited ]
|
`->{5} [ NA ]
->{7} [ YES(?,O(1)) ]
->{4} [ NA ]
|
`->{3} [ NA ]
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [7]
* Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {minus^#(x, 0()) -> c_0(x)}
Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
s(x1) = [1] x1 + [2]
minus^#(x1, x2) = [2] x1 + [2] x2 + [2]
c_0(x1) = [0] x1 + [1]
c_1(x1) = [1] x1 + [5]
* Path {4}: NA
------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {4}->{3}: NA
-----------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {7}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {reverse^#(nil()) -> c_6()}
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_6() = [1]
* Path {8}: inherited
-------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{5}: 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 {8}->{6}: inherited
------------------------
This path is subsumed by the proof of path {8}->{6}->{5}.
* Path {8}->{6}->{5}: 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}->{9}.
* Path {10}->{9}: 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:
{ shuffle^#(add(n, x)) -> c_9(n, shuffle^#(reverse(x)))
, shuffle^#(nil()) -> c_8()
, 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 {12}: YES(?,O(n^1))
------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1, x2) = [1] x1 + [1] x2 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_11) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cons(x1, x2) = [1] x1 + [1] x2 + [4]
concat^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_11(x1, x2) = [0] x1 + [1] x2 + [7]
* Path {12}->{11}: 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(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {}, Uargs(cons) = {},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}, Uargs(less_leaves^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [0] x1 + [0] x2 + [0]
leaf() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_10(x1) = [1] x1 + [0]
c_11(x1, x2) = [0] x1 + [1] x2 + [0]
less_leaves^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {concat^#(leaf(), y) -> c_10(y)}
Weak Rules: {concat^#(cons(u, v), y) -> c_11(u, concat^#(v, y))}
Proof Output:
The following argument positions are usable:
Uargs(cons) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
leaf() = [2]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
concat^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_10(x1) = [1] x1 + [1]
c_11(x1, x2) = [0] x1 + [1] x2 + [3]
* Path {15}: NA
-------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, 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 {15}->{13}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {2}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [2] x1 + [3] x2 + [1]
leaf() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {15}->{14}: NA
-------------------
The usable rules for this path are:
{ concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(reverse) = {},
Uargs(shuffle) = {}, Uargs(concat) = {2}, Uargs(cons) = {2},
Uargs(less_leaves) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(app^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
Uargs(reverse^#) = {}, Uargs(c_7) = {}, Uargs(shuffle^#) = {},
Uargs(c_9) = {}, Uargs(concat^#) = {}, Uargs(c_10) = {},
Uargs(c_11) = {}, Uargs(less_leaves^#) = {1, 2}, Uargs(c_14) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
reverse(x1) = [0] x1 + [0]
shuffle(x1) = [0] x1 + [0]
concat(x1, x2) = [2] x1 + [3] x2 + [1]
leaf() = [3]
cons(x1, x2) = [1] x1 + [1] x2 + [2]
less_leaves(x1, x2) = [0] x1 + [0] x2 + [0]
false() = [0]
true() = [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1, x2) = [0] x1 + [0] x2 + [0]
reverse^#(x1) = [0] x1 + [0]
c_6() = [0]
c_7(x1) = [0] x1 + [0]
shuffle^#(x1) = [0] x1 + [0]
c_8() = [0]
c_9(x1, x2) = [0] x1 + [0] x2 + [0]
concat^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_10(x1) = [0] x1 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
less_leaves^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_12() = [0]
c_13() = [0]
c_14(x1) = [1] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, 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(nil()) -> nil()
, shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))
, less_leaves(x, leaf()) -> false()
, less_leaves(leaf(), cons(w, z)) -> true()
, less_leaves(cons(u, v), cons(w, z)) ->
less_leaves(concat(u, v), concat(w, z))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair2rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, 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(nil()) -> nil()
, shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))
, less_leaves(x, leaf()) -> false()
, less_leaves(leaf(), cons(w, z)) -> true()
, less_leaves(cons(u, v), cons(w, z)) ->
less_leaves(concat(u, v), concat(w, z))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3irc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, 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(nil()) -> nil()
, shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))
, less_leaves(x, leaf()) -> false()
, less_leaves(leaf(), cons(w, z)) -> true()
, less_leaves(cons(u, v), cons(w, z)) ->
less_leaves(concat(u, v), concat(w, z))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool pair3rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, 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(nil()) -> nil()
, shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))
, less_leaves(x, leaf()) -> false()
, less_leaves(leaf(), cons(w, z)) -> true()
, less_leaves(cons(u, v), cons(w, z)) ->
less_leaves(concat(u, v), concat(w, z))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, 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(nil()) -> nil()
, shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))
, less_leaves(x, leaf()) -> false()
, less_leaves(leaf(), cons(w, z)) -> true()
, less_leaves(cons(u, v), cons(w, z)) ->
less_leaves(concat(u, v), concat(w, z))}
StartTerms: basic terms
Strategy: none
Certificate: TIMEOUT
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..Tool tup3irc
| Execution Time | 61.230568ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.53 |
|---|
stdout:
TIMEOUT
We consider the following Problem:
Strict Trs:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)
, quot(0(), s(y)) -> 0()
, quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
, 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(nil()) -> nil()
, shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
, concat(leaf(), y) -> y
, concat(cons(u, v), y) -> cons(u, concat(v, y))
, less_leaves(x, leaf()) -> false()
, less_leaves(leaf(), cons(w, z)) -> true()
, less_leaves(cons(u, v), cons(w, z)) ->
less_leaves(concat(u, v), concat(w, z))}
StartTerms: basic terms
Strategy: innermost
Certificate: TIMEOUT
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
Computation stopped due to timeout after 60.0 seconds
Arrrr..