Tool CaT
stdout:
MAYBE
Problem:
le(0(),Y) -> true()
le(s(X),0()) -> false()
le(s(X),s(Y)) -> le(X,Y)
app(nil(),Y) -> Y
app(cons(N,L),Y) -> cons(N,app(L,Y))
low(N,nil()) -> nil()
low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
iflow(true(),N,cons(M,L)) -> cons(M,low(N,L))
iflow(false(),N,cons(M,L)) -> low(N,L)
high(N,nil()) -> nil()
high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
ifhigh(true(),N,cons(M,L)) -> high(N,L)
ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L))
quicksort(nil()) -> nil()
quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(0(), Y) -> c_0()
, 2: le^#(s(X), 0()) -> c_1()
, 3: le^#(s(X), s(Y)) -> c_2(le^#(X, Y))
, 4: app^#(nil(), Y) -> c_3()
, 5: app^#(cons(N, L), Y) -> c_4(app^#(L, Y))
, 6: low^#(N, nil()) -> c_5()
, 7: low^#(N, cons(M, L)) -> c_6(iflow^#(le(M, N), N, cons(M, L)))
, 8: iflow^#(true(), N, cons(M, L)) -> c_7(low^#(N, L))
, 9: iflow^#(false(), N, cons(M, L)) -> c_8(low^#(N, L))
, 10: high^#(N, nil()) -> c_9()
, 11: high^#(N, cons(M, L)) ->
c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, 12: ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))
, 13: ifhigh^#(false(), N, cons(M, L)) -> c_12(high^#(N, L))
, 14: quicksort^#(nil()) -> c_13()
, 15: quicksort^#(cons(N, L)) ->
c_14(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ inherited ]
|
|->{4} [ NA ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{14} [ YES(?,O(1)) ]
->{11,13,12} [ YES(?,O(n^1)) ]
|
`->{10} [ YES(?,O(n^1)) ]
->{7,9,8} [ YES(?,O(n^1)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_2(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(0(), Y) -> c_0()}
Weak Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
le^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(X), 0()) -> c_1()}
Weak Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [2]
le^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {7,9,8}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {1}, Uargs(iflow^#) = {1},
Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 1] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
0() = [0]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[3 3] [3 3] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
iflow^#(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
high^#(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]
ifhigh^#(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]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We 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:
{ low^#(N, cons(M, L)) -> c_6(iflow^#(le(M, N), N, cons(M, L)))
, iflow^#(false(), N, cons(M, L)) -> c_8(low^#(N, L))
, iflow^#(true(), N, cons(M, L)) -> c_7(low^#(N, L))}
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(le) = {}, Uargs(s) = {}, Uargs(cons) = {}, Uargs(low^#) = {},
Uargs(c_6) = {1}, Uargs(iflow^#) = {}, Uargs(c_7) = {1},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [4 0] x2 + [0]
[0 0] [2 0] [0]
0() = [2]
[0]
true() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [4]
low^#(x1, x2) = [0 0] x1 + [4 1] x2 + [2]
[0 0] [3 0] [0]
c_6(x1) = [1 0] x1 + [1]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [4 0] x3 + [4]
[0 0] [0 4] [0 0] [0]
c_7(x1) = [1 0] x1 + [1]
[0 0] [0]
c_8(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {7,9,8}->{6}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {1}, Uargs(iflow^#) = {1},
Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [3]
0() = [1]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
iflow^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1) = [1 0] x1 + [0]
[0 1] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
high^#(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]
ifhigh^#(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]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We 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: {low^#(N, nil()) -> c_5()}
Weak Rules:
{ low^#(N, cons(M, L)) -> c_6(iflow^#(le(M, N), N, cons(M, L)))
, iflow^#(false(), N, cons(M, L)) -> c_8(low^#(N, L))
, iflow^#(true(), N, cons(M, L)) -> c_7(low^#(N, L))
, 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(le) = {}, Uargs(s) = {}, Uargs(cons) = {}, Uargs(low^#) = {},
Uargs(c_6) = {1}, Uargs(iflow^#) = {}, Uargs(c_7) = {1},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [4]
[0 0] [2 0] [4]
0() = [2]
[0]
true() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
nil() = [2]
[0]
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [4]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [2 0] x2 + [4]
[0 0] [0 0] [4]
c_5() = [1]
[0]
c_6(x1) = [1 0] x1 + [3]
[0 0] [3]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 0] x3 + [0]
[0 0] [0 4] [0 0] [0]
c_7(x1) = [1 0] x1 + [4]
[0 0] [0]
c_8(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {11,13,12}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {1}, Uargs(ifhigh^#) = {1}, Uargs(c_11) = {1},
Uargs(c_12) = {1}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 1] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
0() = [0]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[3 3] [3 3] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
ifhigh^#(x1, x2, x3) = [1 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]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We 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:
{ high^#(N, cons(M, L)) -> c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, ifhigh^#(false(), N, cons(M, L)) -> c_12(high^#(N, L))
, ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))}
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(le) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(high^#) = {}, Uargs(c_10) = {1}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {1}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [4 0] x2 + [0]
[0 0] [2 0] [0]
0() = [2]
[0]
true() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [4]
high^#(x1, x2) = [0 0] x1 + [4 1] x2 + [2]
[0 0] [3 0] [0]
c_10(x1) = [1 0] x1 + [1]
[0 0] [0]
ifhigh^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [4 0] x3 + [4]
[0 0] [0 4] [0 0] [0]
c_11(x1) = [1 0] x1 + [1]
[0 0] [0]
c_12(x1) = [1 0] x1 + [1]
[0 0] [0]
* Path {11,13,12}->{10}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {1}, Uargs(ifhigh^#) = {1}, Uargs(c_11) = {1},
Uargs(c_12) = {1}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [3]
0() = [1]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We 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: {high^#(N, nil()) -> c_9()}
Weak Rules:
{ high^#(N, cons(M, L)) -> c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, ifhigh^#(false(), N, cons(M, L)) -> c_12(high^#(N, L))
, ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))
, 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(le) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(high^#) = {}, Uargs(c_10) = {1}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {1}, Uargs(c_12) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [4]
[0 0] [2 0] [4]
0() = [2]
[0]
true() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
nil() = [2]
[0]
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [4]
[0 0] [0 0] [0]
high^#(x1, x2) = [0 0] x1 + [2 0] x2 + [4]
[0 0] [0 0] [4]
c_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [3]
[0 0] [3]
ifhigh^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 0] x3 + [0]
[0 0] [0 4] [0 0] [0]
c_11(x1) = [1 0] x1 + [4]
[0 0] [0]
c_12(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {14}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3() = [0]
[0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quicksort^#(nil()) -> c_13()}
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_13() = [0]
[1]
* Path {15}: inherited
--------------------
This path is subsumed by the proof of path {15}->{5}->{4}.
* Path {15}->{4}: NA
------------------
The usable rules for this path are:
{ low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
, le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {15}->{5}: inherited
-------------------------
This path is subsumed by the proof of path {15}->{5}->{4}.
* Path {15}->{5}->{4}: NA
-----------------------
The usable rules for this path are:
{ low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
, le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(0(), Y) -> c_0()
, 2: le^#(s(X), 0()) -> c_1()
, 3: le^#(s(X), s(Y)) -> c_2(le^#(X, Y))
, 4: app^#(nil(), Y) -> c_3()
, 5: app^#(cons(N, L), Y) -> c_4(app^#(L, Y))
, 6: low^#(N, nil()) -> c_5()
, 7: low^#(N, cons(M, L)) -> c_6(iflow^#(le(M, N), N, cons(M, L)))
, 8: iflow^#(true(), N, cons(M, L)) -> c_7(low^#(N, L))
, 9: iflow^#(false(), N, cons(M, L)) -> c_8(low^#(N, L))
, 10: high^#(N, nil()) -> c_9()
, 11: high^#(N, cons(M, L)) ->
c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, 12: ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))
, 13: ifhigh^#(false(), N, cons(M, L)) -> c_12(high^#(N, L))
, 14: quicksort^#(nil()) -> c_13()
, 15: quicksort^#(cons(N, L)) ->
c_14(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ inherited ]
|
|->{4} [ NA ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{14} [ YES(?,O(1)) ]
->{11,13,12} [ MAYBE ]
|
`->{10} [ NA ]
->{7,9,8} [ NA ]
|
`->{6} [ NA ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {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_2(x1) = [1] x1 + [7]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(0(), Y) -> c_0()}
Weak Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_0() = [1]
c_2(x1) = [1] x1 + [5]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {le^#(s(X), 0()) -> c_1()}
Weak Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {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_1() = [1]
c_2(x1) = [1] x1 + [7]
* Path {7,9,8}: 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 {7,9,8}->{6}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {1}, Uargs(iflow^#) = {1},
Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
true() = [1]
s(x1) = [1] x1 + [3]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
iflow^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [1] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {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:
{ high^#(N, cons(M, L)) -> c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, ifhigh^#(false(), N, cons(M, L)) -> c_12(high^#(N, L))
, ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))
, 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {1}, Uargs(ifhigh^#) = {1}, Uargs(c_11) = {1},
Uargs(c_12) = {1}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
true() = [1]
s(x1) = [1] x1 + [3]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
ifhigh^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1) = [1] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {14}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_4) = {},
Uargs(low^#) = {}, Uargs(c_6) = {}, Uargs(iflow^#) = {},
Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(high^#) = {},
Uargs(c_10) = {}, Uargs(ifhigh^#) = {}, Uargs(c_11) = {},
Uargs(c_12) = {}, Uargs(quicksort^#) = {}, Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1) = [0] x1 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {quicksort^#(nil()) -> c_13()}
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_13() = [1]
* Path {15}: inherited
--------------------
This path is subsumed by the proof of path {15}->{5}->{4}.
* Path {15}->{4}: NA
------------------
The usable rules for this path are:
{ low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
, le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {15}->{5}: inherited
-------------------------
This path is subsumed by the proof of path {15}->{5}->{4}.
* Path {15}->{5}->{4}: NA
-----------------------
The usable rules for this path are:
{ low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
, le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
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
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(0(), Y) -> c_0()
, 2: le^#(s(X), 0()) -> c_1()
, 3: le^#(s(X), s(Y)) -> c_2(le^#(X, Y))
, 4: app^#(nil(), Y) -> c_3(Y)
, 5: app^#(cons(N, L), Y) -> c_4(N, app^#(L, Y))
, 6: low^#(N, nil()) -> c_5()
, 7: low^#(N, cons(M, L)) -> c_6(iflow^#(le(M, N), N, cons(M, L)))
, 8: iflow^#(true(), N, cons(M, L)) -> c_7(M, low^#(N, L))
, 9: iflow^#(false(), N, cons(M, L)) -> c_8(low^#(N, L))
, 10: high^#(N, nil()) -> c_9()
, 11: high^#(N, cons(M, L)) ->
c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, 12: ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))
, 13: ifhigh^#(false(), N, cons(M, L)) -> c_12(M, high^#(N, L))
, 14: quicksort^#(nil()) -> c_13()
, 15: quicksort^#(cons(N, L)) ->
c_14(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ inherited ]
|
|->{4} [ NA ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{14} [ YES(?,O(1)) ]
->{11,13,12} [ YES(?,O(n^2)) ]
|
`->{10} [ YES(?,O(n^2)) ]
->{7,9,8} [ YES(?,O(n^2)) ]
|
`->{6} [ YES(?,O(n^1)) ]
->{3} [ YES(?,O(n^2)) ]
|
|->{1} [ YES(?,O(n^2)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
false() = [0]
[0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(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_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 6] x1 + [2]
[0 1] [0]
le^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
c_2(x1) = [1 0] x1 + [1]
[0 0] [3]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(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_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(0(), Y) -> c_0()}
Weak Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [0]
[0 1] [2]
le^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
[3 3] [0 1] [0]
c_0() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(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_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(X), 0()) -> c_1()}
Weak Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[2]
s(x1) = [1 0] x1 + [2]
[0 1] [2]
le^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
c_1() = [1]
[0]
c_2(x1) = [1 0] x1 + [3]
[0 0] [0]
* Path {7,9,8}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {1},
Uargs(iflow^#) = {1}, Uargs(c_7) = {2}, Uargs(c_8) = {1},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 1] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
0() = [0]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
[0 1] [0 1] [0]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
[3 3] [3 3] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
iflow^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
high^#(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]
ifhigh^#(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, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We 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, cons(M, L)) -> c_6(iflow^#(le(M, N), N, cons(M, L)))
, iflow^#(false(), N, cons(M, L)) -> c_8(low^#(N, L))
, iflow^#(true(), N, cons(M, L)) -> c_7(M, low^#(N, L))}
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(le) = {}, Uargs(s) = {}, Uargs(cons) = {}, Uargs(low^#) = {},
Uargs(c_6) = {1}, Uargs(iflow^#) = {}, Uargs(c_7) = {2},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 4] x1 + [2 1] x2 + [0]
[0 0] [0 4] [0]
0() = [3]
[2]
true() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 1] [0]
false() = [0]
[0]
cons(x1, x2) = [0 3] x1 + [1 2] x2 + [0]
[0 1] [0 0] [4]
low^#(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [4 0] [0]
c_6(x1) = [1 0] x1 + [1]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [1]
[0 0] [0 4] [0 0] [0]
c_7(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {7,9,8}->{6}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {1},
Uargs(iflow^#) = {1}, Uargs(c_7) = {2}, Uargs(c_8) = {1},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [3]
0() = [1]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(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_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 1] [0]
iflow^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
c_8(x1) = [1 0] x1 + [0]
[0 1] [0]
high^#(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]
ifhigh^#(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, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We 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: {low^#(N, nil()) -> c_5()}
Weak Rules:
{ low^#(N, cons(M, L)) -> c_6(iflow^#(le(M, N), N, cons(M, L)))
, iflow^#(false(), N, cons(M, L)) -> c_8(low^#(N, L))
, iflow^#(true(), N, cons(M, L)) -> c_7(M, low^#(N, L))
, 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(le) = {}, Uargs(s) = {}, Uargs(cons) = {}, Uargs(low^#) = {},
Uargs(c_6) = {1}, Uargs(iflow^#) = {}, Uargs(c_7) = {2},
Uargs(c_8) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [4]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
low^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
c_5() = [0]
[0]
c_6(x1) = [1 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
[0 0] [4 2] [0 0] [0]
c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [1 0] x1 + [0]
[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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {1}, Uargs(ifhigh^#) = {1},
Uargs(c_11) = {1}, Uargs(c_12) = {2}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 1] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
0() = [0]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
false() = [0]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 1] [0 1] [0]
low(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(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_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(x1, x2) = [0 0] x1 + [2 2] x2 + [0]
[3 3] [3 3] [0]
c_9() = [0]
[0]
c_10(x1) = [1 0] x1 + [0]
[0 1] [0]
ifhigh^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [2 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_11(x1) = [1 0] x1 + [0]
[0 1] [0]
c_12(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We 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, cons(M, L)) -> c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, ifhigh^#(false(), N, cons(M, L)) -> c_12(M, high^#(N, L))
, ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))}
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(le) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(high^#) = {}, Uargs(c_10) = {1}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {1}, Uargs(c_12) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
cons(x1, x2) = [0 0] x1 + [0 4] x2 + [1]
[0 0] [0 1] [1]
high^#(x1, x2) = [0 0] x1 + [0 4] x2 + [0]
[4 4] [2 0] [2]
c_10(x1) = [1 0] x1 + [0]
[0 0] [3]
ifhigh^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[0 0] [4 0] [0 0] [0]
c_11(x1) = [1 0] x1 + [0]
[0 0] [0]
c_12(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {1}, Uargs(ifhigh^#) = {1},
Uargs(c_11) = {1}, Uargs(c_12) = {2}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [2 0] x2 + [2]
[0 0] [0 0] [3]
0() = [1]
[0]
true() = [1]
[1]
s(x1) = [1 0] x1 + [1]
[0 0] [0]
false() = [1]
[1]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(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_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We 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, nil()) -> c_9()}
Weak Rules:
{ high^#(N, cons(M, L)) -> c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, ifhigh^#(false(), N, cons(M, L)) -> c_12(M, high^#(N, L))
, ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))
, 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(le) = {}, Uargs(s) = {}, Uargs(cons) = {},
Uargs(high^#) = {}, Uargs(c_10) = {1}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {1}, Uargs(c_12) = {2}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 2] x1 + [0 2] x2 + [0]
[2 2] [2 0] [0]
0() = [2]
[2]
true() = [0]
[0]
s(x1) = [1 2] x1 + [2]
[0 1] [2]
false() = [0]
[0]
nil() = [2]
[0]
cons(x1, x2) = [0 2] x1 + [1 2] x2 + [0]
[0 0] [0 0] [2]
high^#(x1, x2) = [0 0] x1 + [2 0] x2 + [0]
[4 4] [4 4] [0]
c_9() = [1]
[0]
c_10(x1) = [1 0] x1 + [0]
[2 0] [7]
ifhigh^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [2 0] x3 + [0]
[0 0] [4 4] [2 2] [0]
c_11(x1) = [1 0] x1 + [0]
[0 0] [3]
c_12(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [4]
* Path {14}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
false() = [0]
[0]
app(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
nil() = [0]
[0]
cons(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]
iflow(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]
ifhigh(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]
le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
app^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(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_5() = [0]
[0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
iflow^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
high^#(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]
ifhigh^#(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, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
quicksort^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_13() = [0]
[0]
c_14(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quicksort^#(nil()) -> c_13()}
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_13() = [0]
[1]
* Path {15}: inherited
--------------------
This path is subsumed by the proof of path {15}->{5}->{4}.
* Path {15}->{4}: NA
------------------
The usable rules for this path are:
{ low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
, le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {15}->{5}: inherited
-------------------------
This path is subsumed by the proof of path {15}->{5}->{4}.
* Path {15}->{5}->{4}: NA
-----------------------
The usable rules for this path are:
{ low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
, le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: le^#(0(), Y) -> c_0()
, 2: le^#(s(X), 0()) -> c_1()
, 3: le^#(s(X), s(Y)) -> c_2(le^#(X, Y))
, 4: app^#(nil(), Y) -> c_3(Y)
, 5: app^#(cons(N, L), Y) -> c_4(N, app^#(L, Y))
, 6: low^#(N, nil()) -> c_5()
, 7: low^#(N, cons(M, L)) -> c_6(iflow^#(le(M, N), N, cons(M, L)))
, 8: iflow^#(true(), N, cons(M, L)) -> c_7(M, low^#(N, L))
, 9: iflow^#(false(), N, cons(M, L)) -> c_8(low^#(N, L))
, 10: high^#(N, nil()) -> c_9()
, 11: high^#(N, cons(M, L)) ->
c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, 12: ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))
, 13: ifhigh^#(false(), N, cons(M, L)) -> c_12(M, high^#(N, L))
, 14: quicksort^#(nil()) -> c_13()
, 15: quicksort^#(cons(N, L)) ->
c_14(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{15} [ inherited ]
|
|->{4} [ NA ]
|
`->{5} [ inherited ]
|
`->{4} [ NA ]
->{14} [ YES(?,O(1)) ]
->{11,13,12} [ MAYBE ]
|
`->{10} [ NA ]
->{7,9,8} [ NA ]
|
`->{6} [ NA ]
->{3} [ YES(?,O(n^1)) ]
|
|->{1} [ YES(?,O(n^1)) ]
|
`->{2} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {3}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [1] x1 + [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {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_2(x1) = [1] x1 + [7]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(0(), Y) -> c_0()}
Weak Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
le^#(x1, x2) = [3] x1 + [3] x2 + [2]
c_0() = [1]
c_2(x1) = [1] x1 + [5]
* Path {3}->{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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {1}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {le^#(s(X), 0()) -> c_1()}
Weak Rules: {le^#(s(X), s(Y)) -> c_2(le^#(X, Y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_2) = {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_1() = [1]
c_2(x1) = [1] x1 + [7]
* Path {7,9,8}: 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 {7,9,8}->{6}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {1},
Uargs(iflow^#) = {1}, Uargs(c_7) = {2}, Uargs(c_8) = {1},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
true() = [1]
s(x1) = [1] x1 + [3]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [1] x1 + [0]
iflow^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_7(x1, x2) = [0] x1 + [1] x2 + [0]
c_8(x1) = [1] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {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: runtime-complexity with respect to
Rules:
{ high^#(N, cons(M, L)) -> c_10(ifhigh^#(le(M, N), N, cons(M, L)))
, ifhigh^#(false(), N, cons(M, L)) -> c_12(M, high^#(N, L))
, ifhigh^#(true(), N, cons(M, L)) -> c_11(high^#(N, L))
, 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {1}, Uargs(ifhigh^#) = {1},
Uargs(c_11) = {1}, Uargs(c_12) = {2}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [1] x1 + [1] x2 + [3]
0() = [3]
true() = [1]
s(x1) = [1] x1 + [3]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [1] x1 + [0]
ifhigh^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [1] x1 + [0]
c_12(x1, x2) = [0] x1 + [1] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {14}: 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(le) = {}, Uargs(s) = {}, Uargs(app) = {}, Uargs(cons) = {},
Uargs(low) = {}, Uargs(iflow) = {}, Uargs(high) = {},
Uargs(ifhigh) = {}, Uargs(quicksort) = {}, Uargs(le^#) = {},
Uargs(c_2) = {}, Uargs(app^#) = {}, Uargs(c_3) = {},
Uargs(c_4) = {}, Uargs(low^#) = {}, Uargs(c_6) = {},
Uargs(iflow^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
Uargs(high^#) = {}, Uargs(c_10) = {}, Uargs(ifhigh^#) = {},
Uargs(c_11) = {}, Uargs(c_12) = {}, Uargs(quicksort^#) = {},
Uargs(c_14) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
le(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
true() = [0]
s(x1) = [0] x1 + [0]
false() = [0]
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
low(x1, x2) = [0] x1 + [0] x2 + [0]
iflow(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
high(x1, x2) = [0] x1 + [0] x2 + [0]
ifhigh(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
quicksort(x1) = [0] x1 + [0]
le^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [0]
low^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_5() = [0]
c_6(x1) = [0] x1 + [0]
iflow^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_7(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
high^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_9() = [0]
c_10(x1) = [0] x1 + [0]
ifhigh^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_11(x1) = [0] x1 + [0]
c_12(x1, x2) = [0] x1 + [0] x2 + [0]
quicksort^#(x1) = [0] x1 + [0]
c_13() = [0]
c_14(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {quicksort^#(nil()) -> c_13()}
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_13() = [1]
* Path {15}: inherited
--------------------
This path is subsumed by the proof of path {15}->{5}->{4}.
* Path {15}->{4}: NA
------------------
The usable rules for this path are:
{ low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
, le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {15}->{5}: inherited
-------------------------
This path is subsumed by the proof of path {15}->{5}->{4}.
* Path {15}->{5}->{4}: NA
-----------------------
The usable rules for this path are:
{ low(N, nil()) -> nil()
, low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
, high(N, nil()) -> nil()
, high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
, quicksort(nil()) -> nil()
, quicksort(cons(N, L)) ->
app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
, le(0(), Y) -> true()
, le(s(X), 0()) -> false()
, le(s(X), s(Y)) -> le(X, Y)
, app(nil(), Y) -> Y
, app(cons(N, L), Y) -> cons(N, app(L, Y))
, iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
, iflow(false(), N, cons(M, L)) -> low(N, L)
, ifhigh(true(), N, cons(M, L)) -> high(N, L)
, ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
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.