Tool CaT
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.55 |
|---|
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)))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
app(nil(),y) -> y
app(add(n,x),y) -> add(n,app(x,y))
low(n,nil()) -> nil()
low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
if_low(true(),n,add(m,x)) -> add(m,low(n,x))
if_low(false(),n,add(m,x)) -> low(n,x)
high(n,nil()) -> nil()
high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
if_high(true(),n,add(m,x)) -> high(n,x)
if_high(false(),n,add(m,x)) -> add(m,high(n,x))
quicksort(nil()) -> nil()
quicksort(add(n,x)) -> app(quicksort(low(n,x)),add(n,quicksort(high(n,x))))
Proof:
OpenTool IRC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.55 |
|---|
stdout:
MAYBE
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.55 |
|---|
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)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))}
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: le^#(0(), y) -> c_4()
, 6: le^#(s(x), 0()) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: app^#(nil(), y) -> c_7()
, 9: app^#(add(n, x), y) -> c_8(app^#(x, y))
, 10: low^#(n, nil()) -> c_9()
, 11: low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, 12: if_low^#(true(), n, add(m, x)) -> c_11(low^#(n, x))
, 13: if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, 14: high^#(n, nil()) -> c_13()
, 15: high^#(n, add(m, x)) ->
c_14(if_high^#(le(m, n), n, add(m, x)))
, 16: if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))
, 17: if_high^#(false(), n, add(m, x)) -> c_16(high^#(n, x))
, 18: quicksort^#(nil()) -> c_17()
, 19: quicksort^#(add(n, x)) ->
c_18(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ inherited ]
|
|->{8} [ NA ]
|
`->{9} [ inherited ]
|
`->{8} [ NA ]
->{18} [ YES(?,O(1)) ]
->{15,17,16} [ YES(?,O(n^2)) ]
|
`->{14} [ YES(?,O(n^2)) ]
->{11,13,12} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{7} [ YES(?,O(n^2)) ]
|
|->{5} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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}: YES(?,O(n^2))
-----------------------
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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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:
{quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))}
Weak Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 4] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
quot^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [0]
c_3(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y)) -> c_2()}
Weak Rules:
{ quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[4 4] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [1]
[0 0] [2]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [2 1] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {7}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {1}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_6(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {7}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {1}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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: {le^#(0(), y) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_4() = [1]
[0]
c_6(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {7}->{6}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {1}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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: {le^#(s(x), 0()) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_5() = [1]
[0]
c_6(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {11,13,12}: YES(?,O(n^2))
------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {1}, Uargs(if_low^#) = {1},
Uargs(c_11) = {1}, Uargs(c_12) = {1}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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 0] x1 + [0]
[0 1] [1]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
le(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [3]
true() = [0]
[1]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(x1, x2) = [0 1] x1 + [1 3] x2 + [0]
[0 1] [0 0] [0]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
if_low^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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:
{ low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, if_low^#(true(), n, add(m, x)) -> c_11(low^#(n, x))}
Weak Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le) = {}, Uargs(add) = {}, Uargs(low^#) = {},
Uargs(c_10) = {1}, Uargs(if_low^#) = {}, Uargs(c_11) = {1},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[4]
s(x1) = [1 0] x1 + [0]
[0 1] [4]
le(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [2]
[0]
false() = [4]
[0]
add(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
[0 1] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [4 1] x2 + [1]
[0 0] [0 2] [0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [4 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [1 2] x1 + [0]
[0 0] [0]
c_12(x1) = [1 0] x1 + [2]
[0 0] [0]
* Path {11,13,12}->{10}: YES(?,O(n^2))
------------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {1}, Uargs(if_low^#) = {1},
Uargs(c_11) = {1}, Uargs(c_12) = {1}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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() = [1]
[0]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
le(x1, x2) = [2 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
true() = [1]
[1]
false() = [1]
[1]
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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
if_low^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {low^#(n, nil()) -> c_9()}
Weak Rules:
{ low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, if_low^#(true(), n, add(m, x)) -> c_11(low^#(n, x))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le) = {}, Uargs(add) = {}, Uargs(low^#) = {},
Uargs(c_10) = {1}, Uargs(if_low^#) = {}, Uargs(c_11) = {1},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[4]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
le(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[0 1] [0 2] [0]
true() = [0]
[0]
false() = [1]
[0]
nil() = [2]
[0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [2 0] x2 + [0]
[0 4] [6 4] [0]
c_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 2] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 0] x3 + [0]
[1 0] [0 0] [0 0] [3]
c_11(x1) = [1 0] x1 + [1]
[0 0] [3]
c_12(x1) = [1 0] x1 + [2]
[0 0] [4]
* Path {15,17,16}: YES(?,O(n^2))
------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {1}, Uargs(if_high^#) = {1}, Uargs(c_15) = {1},
Uargs(c_16) = {1}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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 0] x1 + [0]
[0 1] [1]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
le(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [3]
true() = [0]
[1]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(x1, x2) = [0 1] x1 + [1 3] x2 + [0]
[0 1] [0 0] [0]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
[3 3] [3 3] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
if_high^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
c_16(x1) = [1 0] x1 + [0]
[0 1] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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:
{ high^#(n, add(m, x)) -> c_14(if_high^#(le(m, n), n, add(m, x)))
, if_high^#(false(), n, add(m, x)) -> c_16(high^#(n, x))
, if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))}
Weak Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le) = {}, Uargs(add) = {}, Uargs(high^#) = {},
Uargs(c_14) = {1}, Uargs(if_high^#) = {}, Uargs(c_15) = {1},
Uargs(c_16) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[4]
s(x1) = [1 0] x1 + [0]
[0 1] [4]
le(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [2]
[0]
false() = [4]
[0]
add(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
[0 1] [0 0] [0]
high^#(x1, x2) = [0 0] x1 + [4 1] x2 + [1]
[0 0] [0 2] [0]
c_14(x1) = [1 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [4 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [1 2] x1 + [0]
[0 0] [0]
c_16(x1) = [1 0] x1 + [2]
[0 0] [0]
* Path {15,17,16}->{14}: YES(?,O(n^2))
------------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {1}, Uargs(if_high^#) = {1}, Uargs(c_15) = {1},
Uargs(c_16) = {1}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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() = [1]
[0]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
le(x1, x2) = [2 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
true() = [1]
[1]
false() = [1]
[1]
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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
if_high^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
c_16(x1) = [1 0] x1 + [0]
[0 1] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {high^#(n, nil()) -> c_13()}
Weak Rules:
{ high^#(n, add(m, x)) -> c_14(if_high^#(le(m, n), n, add(m, x)))
, if_high^#(false(), n, add(m, x)) -> c_16(high^#(n, x))
, if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le) = {}, Uargs(add) = {}, Uargs(high^#) = {},
Uargs(c_14) = {1}, Uargs(if_high^#) = {}, Uargs(c_15) = {1},
Uargs(c_16) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[4]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
le(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[0 1] [0 2] [0]
true() = [0]
[0]
false() = [1]
[0]
nil() = [2]
[0]
add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [0]
high^#(x1, x2) = [0 0] x1 + [2 0] x2 + [0]
[0 4] [6 4] [0]
c_13() = [1]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 2] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 0] x3 + [0]
[1 0] [0 0] [0 0] [3]
c_15(x1) = [1 0] x1 + [1]
[0 0] [3]
c_16(x1) = [1 0] x1 + [2]
[0 0] [4]
* Path {18}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7() = [0]
[0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1) = [0 0] x1 + [0]
[0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1) = [0 0] x1 + [0]
[0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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: {quicksort^#(nil()) -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(quicksort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
quicksort^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_17() = [0]
[1]
* Path {19}: inherited
--------------------
This path is subsumed by the proof of path {19}->{9}->{8}.
* Path {19}->{8}: NA
------------------
The usable rules for this path are:
{ low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))}
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 {19}->{9}: inherited
-------------------------
This path is subsumed by the proof of path {19}->{9}->{8}.
* Path {19}->{9}->{8}: NA
-----------------------
The usable rules for this path are:
{ low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: 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: le^#(0(), y) -> c_4()
, 6: le^#(s(x), 0()) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: app^#(nil(), y) -> c_7()
, 9: app^#(add(n, x), y) -> c_8(app^#(x, y))
, 10: low^#(n, nil()) -> c_9()
, 11: low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, 12: if_low^#(true(), n, add(m, x)) -> c_11(low^#(n, x))
, 13: if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, 14: high^#(n, nil()) -> c_13()
, 15: high^#(n, add(m, x)) ->
c_14(if_high^#(le(m, n), n, add(m, x)))
, 16: if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))
, 17: if_high^#(false(), n, add(m, x)) -> c_16(high^#(n, x))
, 18: quicksort^#(nil()) -> c_17()
, 19: quicksort^#(add(n, x)) ->
c_18(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ inherited ]
|
|->{8} [ NA ]
|
`->{9} [ inherited ]
|
`->{8} [ NA ]
->{18} [ YES(?,O(1)) ]
->{15,17,16} [ NA ]
|
`->{14} [ NA ]
->{11,13,12} [ MAYBE ]
|
`->{10} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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}: YES(?,O(n^1))
-----------------------
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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))}
Weak Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [4]
quot^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_3(x1) = [1] x1 + [3]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {1}, Uargs(c_3) = {1}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y)) -> c_2()}
Weak Rules:
{ quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [1]
s(x1) = [1] x1 + [0]
quot^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
* Path {7}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {1}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {1}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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: {le^#(0(), y) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_4() = [1]
c_6(x1) = [1] x1 + [2]
* Path {7}->{6}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {1}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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: {le^#(s(x), 0()) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5() = [1]
c_6(x1) = [1] x1 + [7]
* Path {11,13,12}: MAYBE
----------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, if_low^#(true(), n, add(m, x)) -> c_11(low^#(n, x))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {11,13,12}->{10}: NA
-------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {1}, Uargs(if_low^#) = {1},
Uargs(c_11) = {1}, Uargs(c_12) = {1}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
le(x1, x2) = [1] x1 + [1] x2 + [3]
true() = [1]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
if_low^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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 {15,17,16}: NA
-------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(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 {15,17,16}->{14}: NA
-------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {1}, Uargs(if_high^#) = {1}, Uargs(c_15) = {1},
Uargs(c_16) = {1}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
le(x1, x2) = [1] x1 + [1] x2 + [3]
true() = [1]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [1] x1 + [0]
if_high^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1) = [1] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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 {18}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
Uargs(quot^#) = {}, Uargs(c_3) = {}, Uargs(le^#) = {},
Uargs(c_6) = {}, Uargs(app^#) = {}, Uargs(c_8) = {},
Uargs(low^#) = {}, Uargs(c_10) = {}, Uargs(if_low^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(high^#) = {},
Uargs(c_14) = {}, Uargs(if_high^#) = {}, Uargs(c_15) = {},
Uargs(c_16) = {}, Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7() = [0]
c_8(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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: {quicksort^#(nil()) -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(quicksort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
quicksort^#(x1) = [1] x1 + [7]
c_17() = [1]
* Path {19}: inherited
--------------------
This path is subsumed by the proof of path {19}->{9}->{8}.
* Path {19}->{8}: NA
------------------
The usable rules for this path are:
{ low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))}
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 {19}->{9}: inherited
-------------------------
This path is subsumed by the proof of path {19}->{9}->{8}.
* Path {19}->{9}->{8}: NA
-----------------------
The usable rules for this path are:
{ low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.55 |
|---|
stdout:
MAYBE
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | MAYBE |
|---|
| Input | AG01 3.55 |
|---|
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)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))}
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: le^#(0(), y) -> c_4()
, 6: le^#(s(x), 0()) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: app^#(nil(), y) -> c_7(y)
, 9: app^#(add(n, x), y) -> c_8(n, app^#(x, y))
, 10: low^#(n, nil()) -> c_9()
, 11: low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, 12: if_low^#(true(), n, add(m, x)) -> c_11(m, low^#(n, x))
, 13: if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, 14: high^#(n, nil()) -> c_13()
, 15: high^#(n, add(m, x)) ->
c_14(if_high^#(le(m, n), n, add(m, x)))
, 16: if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))
, 17: if_high^#(false(), n, add(m, x)) -> c_16(m, high^#(n, x))
, 18: quicksort^#(nil()) -> c_17()
, 19: quicksort^#(add(n, x)) ->
c_18(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ inherited ]
|
|->{8} [ NA ]
|
`->{9} [ inherited ]
|
`->{8} [ NA ]
->{18} [ YES(?,O(1)) ]
->{15,17,16} [ YES(?,O(n^2)) ]
|
`->{14} [ YES(?,O(n^1)) ]
->{11,13,12} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{7} [ YES(?,O(n^2)) ]
|
|->{5} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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}: YES(?,O(n^2))
-----------------------
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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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:
{quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))}
Weak Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 4] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
quot^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[0 0] [0 1] [0]
c_3(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y)) -> c_2()}
Weak Rules:
{ quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[4 4] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [1]
[0 0] [2]
quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [2 1] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {7}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_6(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {7}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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: {le^#(0(), y) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_4() = [1]
[0]
c_6(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {7}->{6}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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: {le^#(s(x), 0()) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_5() = [1]
[0]
c_6(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {11,13,12}: YES(?,O(n^2))
------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {1}, Uargs(if_low^#) = {1}, Uargs(c_11) = {2},
Uargs(c_12) = {1}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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 0] x1 + [0]
[0 1] [1]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
le(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [3]
true() = [0]
[1]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
add(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [0]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [2 1] x2 + [0]
[3 3] [3 3] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
if_low^#(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [2 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_11(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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:
{ low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, if_low^#(true(), n, add(m, x)) -> c_11(m, low^#(n, x))}
Weak Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le) = {}, Uargs(add) = {}, Uargs(low^#) = {},
Uargs(c_10) = {1}, Uargs(if_low^#) = {}, Uargs(c_11) = {2},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [1]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[4 0] [0 0] [0]
true() = [0]
[2]
false() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[0 0] [0 1] [2]
low^#(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [1]
[0 2] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [2]
c_12(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {11,13,12}->{10}: YES(?,O(n^2))
------------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {1}, Uargs(if_low^#) = {1}, Uargs(c_11) = {2},
Uargs(c_12) = {1}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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() = [1]
[0]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
le(x1, x2) = [2 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
true() = [1]
[1]
false() = [1]
[1]
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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
if_low^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
c_12(x1) = [1 0] x1 + [0]
[0 1] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {low^#(n, nil()) -> c_9()}
Weak Rules:
{ low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, if_low^#(true(), n, add(m, x)) -> c_11(m, low^#(n, x))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le) = {}, Uargs(add) = {}, Uargs(low^#) = {},
Uargs(c_10) = {1}, Uargs(if_low^#) = {}, Uargs(c_11) = {2},
Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [1]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
le(x1, x2) = [0 0] x1 + [4 0] x2 + [0]
[4 2] [4 0] [0]
true() = [0]
[0]
false() = [0]
[0]
nil() = [2]
[2]
add(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 1] [4]
low^#(x1, x2) = [0 0] x1 + [2 2] x2 + [0]
[4 0] [2 0] [0]
c_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 2] x3 + [0]
[0 0] [4 4] [0 0] [4]
c_11(x1, x2) = [0 0] x1 + [1 0] x2 + [7]
[0 0] [0 0] [3]
c_12(x1) = [1 0] x1 + [6]
[0 0] [3]
* Path {15,17,16}: YES(?,O(n^2))
------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {1},
Uargs(if_high^#) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {2},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
[1]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
le(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [3]
true() = [0]
[1]
false() = [1]
[1]
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 1] x2 + [0]
[0 1] [0 1] [0]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[3 3] [3 3] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
if_high^#(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
c_16(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ high^#(n, add(m, x)) -> c_14(if_high^#(le(m, n), n, add(m, x)))
, if_high^#(false(), n, add(m, x)) -> c_16(m, high^#(n, x))
, if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))}
Weak Rules:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le) = {}, Uargs(add) = {}, Uargs(high^#) = {},
Uargs(c_14) = {1}, Uargs(if_high^#) = {}, Uargs(c_15) = {1},
Uargs(c_16) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
add(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [1]
high^#(x1, x2) = [0 0] x1 + [4 4] x2 + [0]
[4 4] [1 0] [4]
c_14(x1) = [1 0] x1 + [1]
[0 0] [2]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [4 1] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [1 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
* Path {15,17,16}->{14}: YES(?,O(n^1))
------------------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {1},
Uargs(if_high^#) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {2},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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() = [1]
[0]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
le(x1, x2) = [2 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
true() = [1]
[1]
false() = [1]
[1]
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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 1] [0]
if_high^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [1 0] x1 + [0]
[0 1] [0]
c_16(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {high^#(n, nil()) -> c_13()}
Weak Rules:
{ high^#(n, add(m, x)) -> c_14(if_high^#(le(m, n), n, add(m, x)))
, if_high^#(false(), n, add(m, x)) -> c_16(m, high^#(n, x))
, if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le) = {}, Uargs(add) = {}, Uargs(high^#) = {},
Uargs(c_14) = {1}, Uargs(if_high^#) = {}, Uargs(c_15) = {1},
Uargs(c_16) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [4]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
nil() = [0]
[0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [1 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [1 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
* Path {18}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [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]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_low(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
high(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if_high(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
quicksort(x1) = [0 0] x1 + [0]
[0 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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_9() = [0]
[0]
c_10(x1) = [0 0] x1 + [0]
[0 0] [0]
if_low^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_12(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
if_high^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_15(x1) = [0 0] x1 + [0]
[0 0] [0]
c_16(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_17() = [0]
[0]
c_18(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: {quicksort^#(nil()) -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(quicksort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [2]
[2]
quicksort^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_17() = [0]
[1]
* Path {19}: inherited
--------------------
This path is subsumed by the proof of path {19}->{9}->{8}.
* Path {19}->{8}: NA
------------------
The usable rules for this path are:
{ low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))}
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 {19}->{9}: inherited
-------------------------
This path is subsumed by the proof of path {19}->{9}->{8}.
* Path {19}->{9}->{8}: NA
-----------------------
The usable rules for this path are:
{ low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: 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: le^#(0(), y) -> c_4()
, 6: le^#(s(x), 0()) -> c_5()
, 7: le^#(s(x), s(y)) -> c_6(le^#(x, y))
, 8: app^#(nil(), y) -> c_7(y)
, 9: app^#(add(n, x), y) -> c_8(n, app^#(x, y))
, 10: low^#(n, nil()) -> c_9()
, 11: low^#(n, add(m, x)) -> c_10(if_low^#(le(m, n), n, add(m, x)))
, 12: if_low^#(true(), n, add(m, x)) -> c_11(m, low^#(n, x))
, 13: if_low^#(false(), n, add(m, x)) -> c_12(low^#(n, x))
, 14: high^#(n, nil()) -> c_13()
, 15: high^#(n, add(m, x)) ->
c_14(if_high^#(le(m, n), n, add(m, x)))
, 16: if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))
, 17: if_high^#(false(), n, add(m, x)) -> c_16(m, high^#(n, x))
, 18: quicksort^#(nil()) -> c_17()
, 19: quicksort^#(add(n, x)) ->
c_18(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{19} [ inherited ]
|
|->{8} [ NA ]
|
`->{9} [ inherited ]
|
`->{8} [ NA ]
->{18} [ YES(?,O(1)) ]
->{15,17,16} [ MAYBE ]
|
`->{14} [ NA ]
->{11,13,12} [ NA ]
|
`->{10} [ NA ]
->{7} [ YES(?,O(n^1)) ]
|
|->{5} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {1}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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}: YES(?,O(n^1))
-----------------------
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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))}
Weak Rules:
{ minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [4]
quot^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_3(x1) = [1] x1 + [3]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {1}, Uargs(c_3) = {1},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quot^#(0(), s(y)) -> c_2()}
Weak Rules:
{ quot^#(s(x), s(y)) -> c_3(quot^#(minus(x, y), s(y)))
, minus(x, 0()) -> x
, minus(s(x), s(y)) -> minus(x, y)}
Proof Output:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot^#) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1] x1 + [0] x2 + [0]
0() = [1]
s(x1) = [1] x1 + [0]
quot^#(x1, x2) = [1] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
* Path {7}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_6(x1) = [1] x1 + [7]
* Path {7}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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: {le^#(0(), y) -> c_4()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_4() = [1]
c_6(x1) = [1] x1 + [2]
* Path {7}->{6}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {1}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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: {le^#(s(x), 0()) -> c_5()}
Weak Rules: {le^#(s(x), s(y)) -> c_6(le^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_6) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_5() = [1]
c_6(x1) = [1] x1 + [7]
* Path {11,13,12}: NA
-------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(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 {11,13,12}->{10}: NA
-------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {1}, Uargs(if_low^#) = {1}, Uargs(c_11) = {2},
Uargs(c_12) = {1}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
le(x1, x2) = [1] x1 + [1] x2 + [3]
true() = [1]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
if_low^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [1] x2 + [0]
c_12(x1) = [1] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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 {15,17,16}: MAYBE
----------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(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:
{ high^#(n, add(m, x)) -> c_14(if_high^#(le(m, n), n, add(m, x)))
, if_high^#(false(), n, add(m, x)) -> c_16(m, high^#(n, x))
, if_high^#(true(), n, add(m, x)) -> c_15(high^#(n, x))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
Proof Output:
The input cannot be shown compatible
* Path {15,17,16}->{14}: NA
-------------------------
The usable rules for this path are:
{ le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(quot) = {}, Uargs(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {1},
Uargs(if_high^#) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {2},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [3]
s(x1) = [1] x1 + [3]
quot(x1, x2) = [0] x1 + [0] x2 + [0]
le(x1, x2) = [1] x1 + [1] x2 + [3]
true() = [1]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [1] x1 + [0]
if_high^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [1] x1 + [0]
c_16(x1, x2) = [0] x1 + [1] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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 {18}: 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(le) = {},
Uargs(app) = {}, Uargs(add) = {}, Uargs(low) = {},
Uargs(if_low) = {}, Uargs(high) = {}, Uargs(if_high) = {},
Uargs(quicksort) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_1) = {}, Uargs(quot^#) = {}, Uargs(c_3) = {},
Uargs(le^#) = {}, Uargs(c_6) = {}, Uargs(app^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(low^#) = {},
Uargs(c_10) = {}, Uargs(if_low^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(high^#) = {}, Uargs(c_14) = {},
Uargs(if_high^#) = {}, Uargs(c_15) = {}, Uargs(c_16) = {},
Uargs(quicksort^#) = {}, Uargs(c_18) = {}
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]
le(x1, x2) = [0] x1 + [0] x2 + [0]
true() = [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
if_low(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
if_high(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [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]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
if_low^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1, x2) = [0] x1 + [0] x2 + [0]
c_12(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
if_high^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_15(x1) = [0] x1 + [0]
c_16(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_17() = [0]
c_18(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: {quicksort^#(nil()) -> c_17()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(quicksort^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
nil() = [7]
quicksort^#(x1) = [1] x1 + [7]
c_17() = [1]
* Path {19}: inherited
--------------------
This path is subsumed by the proof of path {19}->{9}->{8}.
* Path {19}->{8}: NA
------------------
The usable rules for this path are:
{ low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))}
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 {19}->{9}: inherited
-------------------------
This path is subsumed by the proof of path {19}->{9}->{8}.
* Path {19}->{9}->{8}: NA
-----------------------
The usable rules for this path are:
{ low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.55 |
|---|
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)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))}
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 3.55 |
|---|
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)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))}
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 3.55 |
|---|
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)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))}
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 3.55 |
|---|
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)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))}
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 3.55 |
|---|
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)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))}
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 | 60.11136ms |
|---|
| Answer | TIMEOUT |
|---|
| Input | AG01 3.55 |
|---|
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)))
, le(0(), y) -> true()
, le(s(x), 0()) -> false()
, le(s(x), s(y)) -> le(x, y)
, app(nil(), y) -> y
, app(add(n, x), y) -> add(n, app(x, y))
, low(n, nil()) -> nil()
, low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
, if_low(true(), n, add(m, x)) -> add(m, low(n, x))
, if_low(false(), n, add(m, x)) -> low(n, x)
, high(n, nil()) -> nil()
, high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
, if_high(true(), n, add(m, x)) -> high(n, x)
, if_high(false(), n, add(m, x)) -> add(m, high(n, x))
, quicksort(nil()) -> nil()
, quicksort(add(n, x)) ->
app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))}
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..